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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap49.html">[Previous Chapter]</a> <a href="chap51.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap50_mj.html">[MathJax on]</a></p>
<p><a id="X81B00B667D2BD022" name="X81B00B667D2BD022"></a></p>
<div class="ChapSects"><a href="chap50.html#X81B00B667D2BD022">50 <span class="Heading">Group Libraries</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X839981CC7D9B671B">50.1 <span class="Heading">Basic Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8489BECB78664847">50.1-1 TrivialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7A7C473D87B31F3B">50.1-2 CyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X81CCC3BF8005A2D7">50.1-3 AbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8778256286E50743">50.1-4 ElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F43050D8587E767">50.1-5 FreeAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X838DE1AB7B3D70FF">50.1-6 DihedralGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X87865686856910E4">50.1-7 QuaternionGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X86E76B3A796BEFA8">50.1-8 ExtraspecialGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E54D3E778E6A53E">50.1-9 <span class="Heading">AlternatingGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X858666F97BD85ABB">50.1-10 <span class="Heading">SymmetricGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X788FA7DE84E0FE6A">50.1-11 MathieuGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8469DBBF82F8E5C3">50.1-12 SuzukiGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X87E5B0F679CA7FE4">50.1-13 ReeGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X8674AAA578FE4AEE">50.2 <span class="Heading">Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X85D607DD82AF3E27">50.2-1 <span class="Heading">GeneralLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7CA3F7BF83992C6B">50.2-2 <span class="Heading">SpecialLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866D4E2B816BDFA5">50.2-3 GeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X82A2AADE805DCDE9">50.2-4 SpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8142A8B07811CA90">50.2-5 <span class="Heading">SymplecticGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7C2051CB7B94CEB1">50.2-6 GeneralOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X78D4EEF27AA2DCFD">50.2-7 SpecialOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8365E0AB8338DA3F">50.2-8 Omega</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X79C3C61A7D83A6D0">50.2-9 GeneralSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7D3779237CB5B49C">50.2-10 SpecialSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F0DBEB880D2D574">50.2-11 ProjectiveGeneralLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X86784EDA80224B74">50.2-12 ProjectiveSpecialLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E471ADE7E095604">50.2-13 ProjectiveGeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7A88FE2B7EF9C804">50.2-14 ProjectiveSpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7DEDE2537B8FFFF5">50.2-15 ProjectiveSymplecticGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7F546F907A37DF15">50.2-16 ProjectiveOmega</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X85B9F2D379616C35">50.3 <span class="Heading">Conjugacy Classes in Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X831789117E93171E">50.3-1 NrConjugacyClassesGL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X817EBD6E841285CD">50.4 <span class="Heading">Constructors for Basic Groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X82676ED5826E9E2E">50.5 <span class="Heading">Selection Functions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7A884ECF813C2026">50.6 <span class="Heading">Finite Perfect Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866A25F882A4E97B">50.6-1 SizesPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7906BBA7818E9415">50.6-2 <span class="Heading">PerfectGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7E1CB2D18085FF9D">50.6-3 PerfectIdentification</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7D68BE547FE5C0F5">50.6-4 NumberPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7FE695DA86A066E1">50.6-5 NumberPerfectLibraryGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X866356A684F6B15E">50.6-6 SizeNumbersPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X845419F07BB92867">50.6-7 <span class="Heading">DisplayInformationPerfectGroups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X875C5BE67BAB7F71">50.6-8 <span class="Heading">More about the Perfect Groups Library</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7873506D873EDB95">50.7 <span class="Heading">Irreducible Maximal Finite Integral Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8693FD647EF3C53B">50.7-1 ImfNumberQQClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8705F64B7E19DDC7">50.7-2 DisplayImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X8604A2167B2E8434">50.7-3 ImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X78935B307B909101">50.7-4 ImfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X84BF34B27CD5E85C">50.7-5 IsomorphismPermGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap50.html#X7CEDB6CE7BAC4518">50.7-6 IsomorphismPermGroupImfGroup</a></span>
</div></div>
</div>
<h3>50 <span class="Heading">Group Libraries</span></h3>
<p>When you start <strong class="pkg">GAP</strong>, it already knows several groups. Currently <strong class="pkg">GAP</strong> initially knows the following groups:</p>
<ul>
<li><p>some basic groups, such as cyclic groups or symmetric groups (see <a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>),</p>
</li>
<li><p>Classical matrix groups (see <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>),</p>
</li>
<li><p>the transitive permutation groups of degree at most 30, provided by the <strong class="pkg">TransGrp</strong> package (see <a href="../../pkg/TransGrp/htm/CHAP001.htm#SECT001"><span class="RefLink">transgrp: Transitive Permutation Groups</span></a>),</p>
</li>
<li><p>a library of groups of small order (see <a href="../../pkg/SmallGrp/doc/chap1.html#X7C16EA1580AC7586"><span class="RefLink">smallgrp: The Small Groups Library</span></a>),</p>
</li>
<li><p>the finite perfect groups of size at most <span class="SimpleMath">10^6</span> (excluding 11 sizes) (see <a href="chap50.html#X7A884ECF813C2026"><span class="RefLink">50.6</span></a>).</p>
</li>
<li><p>the primitive permutation groups of degree <span class="SimpleMath">< 4096</span>, provided by the <strong class="pkg">PrimGrp</strong> package (see <a href="../../pkg/PrimGrp/doc/chap1.html#X7AE00EA7791F2574"><span class="RefLink">primgrp: Primitive Permutation Groups</span></a>),</p>
</li>
<li><p>the irreducible solvable subgroups of <span class="SimpleMath">GL(n,p)</span> for <span class="SimpleMath">n>1</span> and <span class="SimpleMath">p^n < 256</span>, provided by the <strong class="pkg">PrimGrp</strong> package (see <a href="../../pkg/PrimGrp/doc/chap2.html#X82FD673384BF353B"><span class="RefLink">primgrp: Irreducible Solvable Matrix Groups</span></a>),</p>
</li>
<li><p>the irreducible maximal finite integral matrix groups of dimension at most 31 (see <a href="chap50.html#X7873506D873EDB95"><span class="RefLink">50.7</span></a>),</p>
</li>
<li><p>the crystallographic groups of dimension at most 4</p>
</li>
</ul>
<p>There is usually no relation between the groups in the different libraries and a group may occur in different libraries in different incarnations.</p>
<p>Note that a system administrator may choose to install all, or only a few, or even none of the libraries. So some of the libraries mentioned below may not be available on your installation.</p>
<p><strong class="pkg">GAP</strong> might use data libraries that are available to speed up calculations, for example in using a classification to determine that groups must be isomorphic, based on agreement of properties; or to determine maximal subgroups or subgroup maximality. This will be indicated by an info message of level 2 in the info class <code class="code">InfoPerformance</code>. If the calculation is to be independent of such data library use, for example if it is used to verify the data library, functions can be called with the option <code class="code">NoPrecomputedData</code>, to turn these features off. Doing so might cause significantly longer calculations, or even failure of certain calculations.</p>
<p><a id="X839981CC7D9B671B" name="X839981CC7D9B671B"></a></p>
<h4>50.1 <span class="Heading">Basic Groups</span></h4>
<p>There are several infinite families of groups which are parametrized by numbers. <strong class="pkg">GAP</strong> provides various functions to construct these groups. The functions always permit (but do not require) one to indicate a filter (see <a href="chap13.html#X84EFA4C07D4277BB"><span class="RefLink">13.2</span></a>), for example <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) or <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>), in which the group shall be constructed. There always is a default filter corresponding to a "natural" way to describe the group in question. Note that not every group can be constructed in every filter, there may be theoretical restrictions (<code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>) only works for solvable groups) or methods may be available only for a few filters.</p>
<p>Certain filters may admit additional hints. For example, groups constructed in <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) may be constructed over a specified field, which can be given as second argument of the function that constructs the group; The default field is <code class="func">Rationals</code> (<a href="chap17.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>).</p>
<p><a id="X8489BECB78664847" name="X8489BECB78664847"></a></p>
<h5>50.1-1 TrivialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialGroup</code>( [<var class="Arg">filter</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a trivial group in the category given by the filter <var class="Arg">filter</var>. If <var class="Arg">filter</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TrivialGroup();</span>
<pc group of size 1 with 0 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">TrivialGroup( IsPermGroup );</span>
Group(())
</pre></div>
<p><a id="X7A7C473D87B31F3B" name="X7A7C473D87B31F3B"></a></p>
<h5>50.1-2 CyclicGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclicGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the cyclic group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>), unless <var class="Arg">n</var> equals <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), in which case the default filter is switched to <code class="func">IsFpGroup</code> (<a href="chap47.html#X850B9DF17D90C3A2"><span class="RefLink">47.1-2</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicGroup(12);</span>
<pc group of size 12 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicGroup(infinity);</span>
<free group on the generators [ a ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">CyclicGroup(IsPermGroup,12);</span>
Group([ (1,2,3,4,5,6,7,8,9,10,11,12) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">matgrp1:= CyclicGroup( IsMatrixGroup, 12 );</span>
<matrix group of size 12 with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FieldOfMatrixGroup( matgrp1 );</span>
Rationals
<span class="GAPprompt">gap></span> <span class="GAPinput">matgrp2:= CyclicGroup( IsMatrixGroup, GF(2), 12 );</span>
<matrix group of size 12 with 1 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FieldOfMatrixGroup( matgrp2 );</span>
GF(2)
</pre></div>
<p><a id="X81CCC3BF8005A2D7" name="X81CCC3BF8005A2D7"></a></p>
<h5>50.1-3 AbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">ints</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs an abelian group in the category given by the filter <var class="Arg">filt</var> which is of isomorphism type <span class="SimpleMath">C_{<var class="Arg">ints</var>[1]} × C_{<var class="Arg">ints</var>[2]} × ... × C_{<var class="Arg">ints</var>[n]}</span>, where <var class="Arg">ints</var> must be a list of non-negative integers or <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>); for the latter value or 0, <span class="SimpleMath">C_{<var class="Arg">ints</var>[i]}</span> is taken as an infinite cyclic group, otherwise as a cyclic group of order <var class="Arg">ints</var>[i]. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>), unless any 0 or <code class="code">infinity</code> is contained in <var class="Arg">ints</var>, in which the default filter is switched to <code class="func">IsFpGroup</code> (<a href="chap47.html#X850B9DF17D90C3A2"><span class="RefLink">47.1-2</span></a>). The generators of the group returned are the elements corresponding to the factors <span class="SimpleMath">C_{<var class="Arg">ints</var>[i]}</span> and hence the integers in <var class="Arg">ints</var>. For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianGroup([1,2,3]);</span>
<pc group of size 6 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AbelianGroup([0,3]);</span>
<fp group of size infinity on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(G);</span>
[ 0, 3 ]
</pre></div>
<p><a id="X8778256286E50743" name="X8778256286E50743"></a></p>
<h5>50.1-4 ElementaryAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementaryAbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the elementary abelian group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementaryAbelianGroup(8192);</span>
<pc group of size 8192 with 13 generators>
</pre></div>
<p><a id="X7F43050D8587E767" name="X7F43050D8587E767"></a></p>
<h5>50.1-5 FreeAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeAbelianGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the free abelian group of rank <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsFpGroup</code> (<a href="chap47.html#X850B9DF17D90C3A2"><span class="RefLink">47.1-2</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeAbelianGroup(4);</span>
<fp group of size infinity on the generators [ f1, f2, f3, f4 ]>
</pre></div>
<p><a id="X838DE1AB7B3D70FF" name="X838DE1AB7B3D70FF"></a></p>
<h5>50.1-6 DihedralGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DihedralGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the dihedral group of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>), unless <var class="Arg">n</var> equals <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>), in which case the default filter is switched to <code class="func">IsFpGroup</code> (<a href="chap47.html#X850B9DF17D90C3A2"><span class="RefLink">47.1-2</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DihedralGroup(8);</span>
<pc group of size 8 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DihedralGroup( IsPermGroup, 8 );</span>
Group([ (1,2,3,4), (2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">DihedralGroup(infinity);</span>
<fp group of size infinity on the generators [ r, s ]>
</pre></div>
<p><a id="X87865686856910E4" name="X87865686856910E4"></a></p>
<h5>50.1-7 QuaternionGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuaternionGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DicyclicGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the generalized quaternion group (or dicyclic group) of size <var class="Arg">n</var> in the category given by the filter <var class="Arg">filt</var>. Here, <var class="Arg">n</var> is a multiple of 4. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). Methods are also available for permutation and matrix groups (of minimal degree and minimal dimension in coprime characteristic).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuaternionGroup(32);</span>
<pc group of size 32 with 5 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=QuaternionGroup(IsMatrixGroup,CF(16),32);</span>
Group([ [ [ 0, 1 ], [ -1, 0 ] ], [ [ E(16), 0 ], [ 0, -E(16)^7 ] ] ])
</pre></div>
<p><a id="X86E76B3A796BEFA8" name="X86E76B3A796BEFA8"></a></p>
<h5>50.1-8 ExtraspecialGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtraspecialGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">order</var>, <var class="Arg">exp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">order</var> be of the form <span class="SimpleMath">p^{2n+1}</span>, for a prime integer <span class="SimpleMath">p</span> and a positive integer <span class="SimpleMath">n</span>. <code class="func">ExtraspecialGroup</code> returns the extraspecial group of order <var class="Arg">order</var> that is determined by <var class="Arg">exp</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <span class="SimpleMath">p</span> is odd then admissible values of <var class="Arg">exp</var> are the exponent of the group (either <span class="SimpleMath">p</span> or <span class="SimpleMath">p^2</span>) or one of <code class="code">'+'</code>, <code class="code">"+"</code>, <code class="code">'-'</code>, <code class="code">"-"</code>. For <span class="SimpleMath">p = 2</span>, only the above plus or minus signs are admissible.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPcGroup</code> (<a href="chap46.html#X7D1F506D7830B1D9"><span class="RefLink">46.3-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 27, 3 );</span>
<pc group of size 27 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 27, '+' );</span>
<pc group of size 27 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtraspecialGroup( 8, "-" );</span>
<pc group of size 8 with 3 generators>
</pre></div>
<p><a id="X7E54D3E778E6A53E" name="X7E54D3E778E6A53E"></a></p>
<h5>50.1-9 <span class="Heading">AlternatingGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlternatingGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlternatingGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">dom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the alternating group of degree <var class="Arg">deg</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). In the second version, the function constructs the alternating group on the points given in the set <var class="Arg">dom</var> which must be a set of positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AlternatingGroup(5);</span>
Alt( [ 1 .. 5 ] )
</pre></div>
<p><a id="X858666F97BD85ABB" name="X858666F97BD85ABB"></a></p>
<h5>50.1-10 <span class="Heading">SymmetricGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">deg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">dom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the symmetric group of degree <var class="Arg">deg</var> in the category given by the filter <var class="Arg">filt</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>). In the second version, the function constructs the symmetric group on the points given in the set <var class="Arg">dom</var> which must be a set of positive integers.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SymmetricGroup(10);</span>
Sym( [ 1 .. 10 ] )
</pre></div>
<p>Note that permutation groups provide special treatment of symmetric and alternating groups, see <a href="chap43.html#X834208CD7C2956A3"><span class="RefLink">43.4</span></a>.</p>
<p><a id="X788FA7DE84E0FE6A" name="X788FA7DE84E0FE6A"></a></p>
<h5>50.1-11 MathieuGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MathieuGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">degree</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the Mathieu group of degree <var class="Arg">degree</var> in the category given by the filter <var class="Arg">filt</var>, where <var class="Arg">degree</var> must be in the set <span class="SimpleMath">{ 9, 10, 11, 12, 21, 22, 23, 24 }</span>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>). For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MathieuGroup( 11 );</span>
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])
</pre></div>
<p><a id="X8469DBBF82F8E5C3" name="X8469DBBF82F8E5C3"></a></p>
<h5>50.1-12 SuzukiGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SuzukiGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sz</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs a group isomorphic to the Suzuki group Sz( <var class="Arg">q</var> ) over the field with <var class="Arg">q</var> elements, where <var class="Arg">q</var> is a non-square power of <span class="SimpleMath">2</span>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the Suzuki group itself. For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SuzukiGroup( 32 );</span>
Sz(32)
</pre></div>
<p><a id="X87E5B0F679CA7FE4" name="X87E5B0F679CA7FE4"></a></p>
<h5>50.1-13 ReeGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReeGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ree</code>( [<var class="Arg">filt</var>, ]<var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Constructs a group isomorphic to the Ree group <span class="SimpleMath">^2G_2(q)</span> where <span class="SimpleMath">q = 3^{1+2m}</span> for <span class="SimpleMath">m</span> a non-negative integer.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>) and the generating matrices are based on <a href="chapBib.html#biBKLM01">[KLM01]</a>. (No particular choice of a generating set is guaranteed.) For more information on possible values of <var class="Arg">filt</var> see section (<a href="chap50.html#X839981CC7D9B671B"><span class="RefLink">50.1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReeGroup( 27 );</span>
Ree(27)
</pre></div>
<p><a id="X8674AAA578FE4AEE" name="X8674AAA578FE4AEE"></a></p>
<h4>50.2 <span class="Heading">Classical Groups</span></h4>
<p>The following functions return classical groups.</p>
<p>For the linear, symplectic, and unitary groups (the latter in dimension at least <span class="SimpleMath">3</span>), the generators are taken from <a href="chapBib.html#biBTay87">[Tay87]</a>. For the unitary groups in dimension <span class="SimpleMath">2</span>, the isomorphism of SU<span class="SimpleMath">(2,q)</span> and SL<span class="SimpleMath">(2,q)</span> is used, see for example <a href="chapBib.html#biBHup67">[Hup67]</a>.</p>
<p>The generators of the general and special orthogonal groups are taken from <a href="chapBib.html#biBIshibashiEarnest94">[IE94]</a> and <a href="chapBib.html#biBKleidmanLiebeck90">[KL90]</a>, except that the generators of the groups in odd dimension in even characteristic are constructed via the isomorphism to a symplectic group, see for example <a href="chapBib.html#biBCar72a">[Car72]</a>.</p>
<p>The generators of the groups <span class="SimpleMath">Ω^ϵ(d, q)</span> are taken from <a href="chapBib.html#biBRylandsTalor98">[RT98]</a>, except that in odd dimension and even characteristic, the generators of SO<span class="SimpleMath">(d, q)</span> are taken for <span class="SimpleMath">Ω(d, q)</span>. Note that the generators claimed in <a href="chapBib.html#biBRylandsTalor98">[RT98, Section 4.5 and 4.6]</a> do not describe orthogonal groups, one would have to transpose these matrices in order to get groups that respect the required forms. The matrices from <a href="chapBib.html#biBRylandsTalor98">[RT98]</a> generate groups of the right isomorphism types but not orthogonal groups, except in the case <span class="SimpleMath">(d,q) = (5,2)</span>, where the matrices from <a href="chapBib.html#biBRylandsTalor98">[RT98]</a> generate the simple group <span class="SimpleMath">S_4(2)'</span> and not the group <span class="SimpleMath">S_4(2)</span>.</p>
<p>The generators for the semilinear groups are constructed from the generators of the corresponding linear groups plus one additional generator that describes the action of the group of field automorphisms; for prime integers <span class="SimpleMath">p</span> and positive integers <span class="SimpleMath">f</span>, this yields the matrix groups <span class="SimpleMath">Gamma</span>L<span class="SimpleMath">(d, p^f)</span> and <span class="SimpleMath">Sigma</span>L<span class="SimpleMath">(d, p^f)</span> as groups of <span class="SimpleMath">d f × df</span> matrices over the field with <span class="SimpleMath">p</span> elements.</p>
<p>For symplectic and orthogonal matrix groups returned by the functions described below, the invariant bilinear form is stored as the value of the attribute <code class="func">InvariantBilinearForm</code> (<a href="chap44.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>). Analogously, the invariant sesquilinear form defining the unitary groups is stored as the value of the attribute <code class="func">InvariantSesquilinearForm</code> (<a href="chap44.html#X82F22079852130C9"><span class="RefLink">44.5-3</span></a>)). The defining quadratic form of orthogonal groups is stored as the value of the attribute <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>).</p>
<p>Note that due to the different sources for the generators, the invariant forms for the groups <span class="SimpleMath">Ω(e,d,q)</span> are in general different from the forms for SO<span class="SimpleMath">(e,d,q)</span> and GO<span class="SimpleMath">(e,d,q)</span>.</p>
<p><a id="X85D607DD82AF3E27" name="X85D607DD82AF3E27"></a></p>
<h5>50.2-1 <span class="Heading">GeneralLinearGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first two forms construct a group isomorphic to the general linear group GL( <var class="Arg">d</var>, <var class="Arg">R</var> ) of all <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices that are invertible over the ring <var class="Arg">R</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The third and the fourth form construct the general linear group over the finite field with <var class="Arg">q</var> elements.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general linear group as a matrix group in its natural action (see also <code class="func">IsNaturalGL</code> (<a href="chap44.html#X86F9A27D7AFAEB5A"><span class="RefLink">44.4-2</span></a>), <code class="func">IsNaturalGLnZ</code> (<a href="chap44.html#X86F9CC1E7DB97CB6"><span class="RefLink">44.6-4</span></a>)).</p>
<p>Currently supported rings <var class="Arg">R</var> are finite fields, the ring <code class="func">Integers</code> (<a href="chap14.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>), and residue class rings <code class="code">Integers mod <var class="Arg">m</var></code>, see <a href="chap14.html#X864BF040862409FC"><span class="RefLink">14.5</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(4,3);</span>
GL(4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(2,Integers);</span>
GL(2,Integers)
<span class="GAPprompt">gap></span> <span class="GAPinput">GL(3,Integers mod 12);</span>
GL(3,Z/12Z)
</pre></div>
<p>Using the <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>) operation it is possible to obtain the corresponding projective groups in a permutation action:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=GL(4,3);;Size(g);</span>
24261120
<span class="GAPprompt">gap></span> <span class="GAPinput">pgl:=Action(g,Orbit(g,Z(3)^0*[1,0,0,0],OnLines),OnLines);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(pgl);</span>
12130560
</pre></div>
<p>If you are interested only in the projective group as a permutation group and not in the correspondence between its moved points and the points in the projective space, you can also use <code class="func">PGL</code> (<a href="chap50.html#X7F0DBEB880D2D574"><span class="RefLink">50.2-11</span></a>).</p>
<p><a id="X7CA3F7BF83992C6B" name="X7CA3F7BF83992C6B"></a></p>
<h5>50.2-2 <span class="Heading">SpecialLinearGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first two forms construct a group isomorphic to the special linear group SL( <var class="Arg">d</var>, <var class="Arg">R</var> ) of all those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the ring <var class="Arg">R</var> whose determinant is the identity of <var class="Arg">R</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The third and the fourth form construct the special linear group over the finite field with <var class="Arg">q</var> elements.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special linear group as a matrix group in its natural action (see also <code class="func">IsNaturalSL</code> (<a href="chap44.html#X84134F08781EB943"><span class="RefLink">44.4-4</span></a>), <code class="func">IsNaturalSLnZ</code> (<a href="chap44.html#X7B0E70127F5D2EAF"><span class="RefLink">44.6-5</span></a>)).</p>
<p>Currently supported rings <var class="Arg">R</var> are finite fields, the ring <code class="func">Integers</code> (<a href="chap14.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>), and residue class rings <code class="code">Integers mod <var class="Arg">m</var></code>, see <a href="chap14.html#X864BF040862409FC"><span class="RefLink">14.5</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialLinearGroup(2,2);</span>
SL(2,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">SL(3,Integers);</span>
SL(3,Integers)
<span class="GAPprompt">gap></span> <span class="GAPinput">SL(4,Integers mod 4);</span>
SL(4,Z/4Z)
</pre></div>
<p><a id="X866D4E2B816BDFA5" name="X866D4E2B816BDFA5"></a></p>
<h5>50.2-3 GeneralUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the general unitary group GU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general unitary group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralUnitaryGroup( 3, 5 );</span>
GU(3,5)
</pre></div>
<p><a id="X82A2AADE805DCDE9" name="X82A2AADE805DCDE9"></a></p>
<h5>50.2-4 SpecialUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the special unitary group GU(<var class="Arg">d</var>, <var class="Arg">q</var>) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements whose determinant is the identity of the field and that respect a fixed nondegenerate sesquilinear form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special unitary group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialUnitaryGroup( 3, 5 );</span>
SU(3,5)
</pre></div>
<p><a id="X8142A8B07811CA90" name="X8142A8B07811CA90"></a></p>
<h5>50.2-5 <span class="Heading">SymplecticGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">ring</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the symplectic group Sp( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements (respectively the ring <var class="Arg">ring</var>) that respect a fixed nondegenerate symplectic form, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the symplectic group itself.</p>
<p>At the moment finite fields or residue class rings <code class="code">Integers mod <var class="Arg">q</var></code>, with <var class="Arg">q</var> an odd prime power are supported.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SymplecticGroup( 4, 2 );</span>
Sp(4,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymplecticGroup(6,Integers mod 9);</span>
Sp(6,Z/9Z)
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span>
95928796265538862080
</pre></div>
<p><a id="X7C2051CB7B94CEB1" name="X7C2051CB7B94CEB1"></a></p>
<h5>50.2-6 GeneralOrthogonalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralOrthogonalGroup</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GO</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the general orthogonal group GO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a non-singular quadratic form (see <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>)) specified by <var class="Arg">e</var>, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>The value of <var class="Arg">e</var> must be <span class="SimpleMath">0</span> for odd <var class="Arg">d</var> (and can optionally be omitted in this case), respectively one of <span class="SimpleMath">1</span> or <span class="SimpleMath">-1</span> for even <var class="Arg">d</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the general orthogonal group itself.</p>
<p>Note that in <a href="chapBib.html#biBKleidmanLiebeck90">[KL90]</a>, GO is defined as the stabilizer <span class="SimpleMath">∆(V, F, κ)</span> of the quadratic form, up to scalars, whereas our GO is called <span class="SimpleMath">I(V, F, κ)</span> there.</p>
<p><a id="X78D4EEF27AA2DCFD" name="X78D4EEF27AA2DCFD"></a></p>
<h5>50.2-7 SpecialOrthogonalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialOrthogonalGroup</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SO</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SpecialOrthogonalGroup</code> returns a group isomorphic to the special orthogonal group SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), which is the subgroup of all those matrices in the general orthogonal group (see <code class="func">GeneralOrthogonalGroup</code> (<a href="chap50.html#X7C2051CB7B94CEB1"><span class="RefLink">50.2-6</span></a>)) that have determinant one, in the category given by the filter <var class="Arg">filt</var>. (The index of SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) in GO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) is <span class="SimpleMath">2</span> if <var class="Arg">q</var> is odd, and <span class="SimpleMath">1</span> if <var class="Arg">q</var> is even.) Also interesting is the group Omega( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), see <code class="func">Omega</code> (<a href="chap50.html#X8365E0AB8338DA3F"><span class="RefLink">50.2-8</span></a>), which is always of index <span class="SimpleMath">2</span> in SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ).</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the special orthogonal group itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralOrthogonalGroup( 3, 7 );</span>
GO(0,3,7)
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneralOrthogonalGroup( -1, 4, 3 );</span>
GO(-1,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialOrthogonalGroup( 1, 4, 4 );</span>
GO(+1,4,4)
</pre></div>
<p><a id="X8365E0AB8338DA3F" name="X8365E0AB8338DA3F"></a></p>
<h5>50.2-8 Omega</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Omega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>constructs a group isomorphic to the group <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a non-singular quadratic form (see <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>)) specified by <var class="Arg">e</var>, and that have square spinor norm in odd characteristic or Dickson invariant <span class="SimpleMath">0</span> in even characteristic, respectively, in the category given by the filter <var class="Arg">filt</var>. For odd <var class="Arg">q</var>, this group has always index two in the corresponding special orthogonal group, which will be conjugate in <span class="SimpleMath">GL(d,q)</span> to the group returned by SO( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), see <code class="func">SpecialOrthogonalGroup</code> (<a href="chap50.html#X78D4EEF27AA2DCFD"><span class="RefLink">50.2-7</span></a>), but may fix a different form (see <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>).</p>
<p>The value of <var class="Arg">e</var> must be <span class="SimpleMath">0</span> for odd <var class="Arg">d</var> (and can optionally be omitted in this case), respectively one of <span class="SimpleMath">1</span> or <span class="SimpleMath">-1</span> for even <var class="Arg">d</var>. If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group is the group <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( 3, 5 ); StructureDescription( g );</span>
Omega(0,3,5)
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( 1, 4, 4 ); StructureDescription( g );</span>
Omega(+1,4,4)
"A5 x A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Omega( -1, 4, 3 ); StructureDescription( g );</span>
Omega(-1,4,3)
"A6"
</pre></div>
<p><a id="X79C3C61A7D83A6D0" name="X79C3C61A7D83A6D0"></a></p>
<h5>50.2-9 GeneralSemilinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralSemilinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GammaL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">GeneralSemilinearGroup</code> returns a group isomorphic to the general semilinear group <span class="SimpleMath">Γ</span>L( <var class="Arg">d</var>, <var class="Arg">q</var> ) of semilinear mappings of the vector space <code class="code">GF( </code><var class="Arg">q</var><code class="code"> )^</code><var class="Arg">d</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group consists of matrices of dimension <var class="Arg">d</var> <span class="SimpleMath">f</span> over the field with <span class="SimpleMath">p</span> elements, where <var class="Arg">q</var> <span class="SimpleMath">= p^f</span>, for a prime integer <span class="SimpleMath">p</span>.</p>
<p><a id="X7D3779237CB5B49C" name="X7D3779237CB5B49C"></a></p>
<h5>50.2-10 SpecialSemilinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SpecialSemilinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SigmaL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SpecialSemilinearGroup</code> returns a group isomorphic to the special semilinear group <span class="SimpleMath">Σ</span>L( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those semilinear mappings of the vector space <code class="code">GF( </code><var class="Arg">q</var><code class="code"> )^</code><var class="Arg">d</var> (see <code class="func">GeneralSemilinearGroup</code> (<a href="chap50.html#X79C3C61A7D83A6D0"><span class="RefLink">50.2-9</span></a>)) whose linear part has determinant one.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsMatrixGroup</code> (<a href="chap44.html#X7E6093FF85F1C3A1"><span class="RefLink">44.1-1</span></a>), and the returned group consists of matrices of dimension <var class="Arg">d</var> <span class="SimpleMath">f</span> over the field with <span class="SimpleMath">p</span> elements, where <var class="Arg">q</var> <span class="SimpleMath">= p^f</span>, for a prime integer <span class="SimpleMath">p</span>.</p>
<p><a id="X7F0DBEB880D2D574" name="X7F0DBEB880D2D574"></a></p>
<h5>50.2-11 ProjectiveGeneralLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveGeneralLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PGL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective general linear group PGL( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X86784EDA80224B74" name="X86784EDA80224B74"></a></p>
<h5>50.2-12 ProjectiveSpecialLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSpecialLinearGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSL</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective special linear group PSL( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements whose determinant is the identity of the field, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7E471ADE7E095604" name="X7E471ADE7E095604"></a></p>
<h5>50.2-13 ProjectiveGeneralUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveGeneralUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PGU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective general unitary group PGU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7A88FE2B7EF9C804" name="X7A88FE2B7EF9C804"></a></p>
<h5>50.2-14 ProjectiveSpecialUnitaryGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSpecialUnitaryGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSU</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective special unitary group PSU( <var class="Arg">d</var>, <var class="Arg">q</var> ) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <span class="SimpleMath"><var class="Arg">q</var>^2</span> elements that respect a fixed nondegenerate sesquilinear form and have determinant 1, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7DEDE2537B8FFFF5" name="X7DEDE2537B8FFFF5"></a></p>
<h5>50.2-15 ProjectiveSymplecticGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveSymplecticGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSP</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSp</code>( [<var class="Arg">filt</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective symplectic group PSp(<var class="Arg">d</var>,<var class="Arg">q</var>) of those <span class="SimpleMath"><var class="Arg">d</var> × <var class="Arg">d</var></span> matrices over the field with <var class="Arg">q</var> elements that respect a fixed nondegenerate symplectic form, modulo the centre, in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X7F546F907A37DF15" name="X7F546F907A37DF15"></a></p>
<h5>50.2-16 ProjectiveOmega</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveOmega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ POmega</code>( [<var class="Arg">filt</var>, ][<var class="Arg">e</var>, ]<var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a group isomorphic to the projective group P<span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ) of <span class="SimpleMath">Ω</span>( <var class="Arg">e</var>, <var class="Arg">d</var>, <var class="Arg">q</var> ), modulo the centre (see <code class="func">Omega</code> (<a href="chap50.html#X8365E0AB8338DA3F"><span class="RefLink">50.2-8</span></a>)), in the category given by the filter <var class="Arg">filt</var>.</p>
<p>If <var class="Arg">filt</var> is not given it defaults to <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>), and the returned group is the action on lines of the underlying vector space.</p>
<p><a id="X85B9F2D379616C35" name="X85B9F2D379616C35"></a></p>
<h4>50.3 <span class="Heading">Conjugacy Classes in Classical Groups</span></h4>
<p>For general and special linear groups (see <code class="func">GeneralLinearGroup</code> (<a href="chap50.html#X85D607DD82AF3E27"><span class="RefLink">50.2-1</span></a>) and <code class="func">SpecialLinearGroup</code> (<a href="chap50.html#X7CA3F7BF83992C6B"><span class="RefLink">50.2-2</span></a>)) <strong class="pkg">GAP</strong> has an efficient method to generate representatives of the conjugacy classes. This uses results from linear algebra on normal forms of matrices. If you know how to do this for other types of classical groups, please, tell us.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := SL(4,9);</span>
SL(4,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses(g);</span>
861
<span class="GAPprompt">gap></span> <span class="GAPinput">cl := ConjugacyClasses(g);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(cl);</span>
861
</pre></div>
<p><a id="X831789117E93171E" name="X831789117E93171E"></a></p>
<h5>50.3-1 NrConjugacyClassesGL</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesGL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesGU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPGL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPGU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPSL</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesPSU</code>( <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSLIsogeneous</code>( <var class="Arg">n</var>, <var class="Arg">q</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrConjugacyClassesSUIsogeneous</code>( <var class="Arg">n</var>, <var class="Arg">q</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first of these functions compute for given positive integer <var class="Arg">n</var> and prime power <var class="Arg">q</var> the number of conjugacy classes in the classical groups GL( <var class="Arg">n</var>, <var class="Arg">q</var> ), GU( <var class="Arg">n</var>, <var class="Arg">q</var> ), SL( <var class="Arg">n</var>, <var class="Arg">q</var> ), SU( <var class="Arg">n</var>, <var class="Arg">q</var> ), PGL( <var class="Arg">n</var>, <var class="Arg">q</var> ), PGU( <var class="Arg">n</var>, <var class="Arg">q</var> ), PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ), PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ), respectively. (See also <code class="func">ConjugacyClasses</code> (<a href="chap39.html#X871B570284BBA685"><span class="RefLink">39.10-2</span></a>) and Section <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>.)</p>
<p>For each divisor <var class="Arg">f</var> of <var class="Arg">n</var> there is a group of Lie type with the same order as SL( <var class="Arg">n</var>, <var class="Arg">q</var> ), such that its derived subgroup modulo its center is isomorphic to PSL( <var class="Arg">n</var>, <var class="Arg">q</var> ). The various such groups with fixed <var class="Arg">n</var> and <var class="Arg">q</var> are called <em>isogeneous</em>. (Depending on congruence conditions on <var class="Arg">q</var> and <var class="Arg">n</var> several of these groups may actually be isomorphic.) The function <code class="func">NrConjugacyClassesSLIsogeneous</code> computes the number of conjugacy classes in this group. The extreme cases <var class="Arg">f</var> <span class="SimpleMath">= 1</span> and <var class="Arg">f</var> <span class="SimpleMath">= n</span> lead to the groups SL( <var class="Arg">n</var>, <var class="Arg">q</var> ) and PGL( <var class="Arg">n</var>, <var class="Arg">q</var> ), respectively.</p>
<p>The function <code class="func">NrConjugacyClassesSUIsogeneous</code> is the analogous one for the corresponding unitary groups.</p>
<p>The formulae for the number of conjugacy classes are taken from <a href="chapBib.html#biBMac81">[Mac81]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClassesGL(24,27);</span>
22528399544939174406067288580609952
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClassesPSU(19,17);</span>
15052300411163848367708
<span class="GAPprompt">gap></span> <span class="GAPinput">NrConjugacyClasses(SL(16,16));</span>
1229782938228219920
</pre></div>
<p><a id="X817EBD6E841285CD" name="X817EBD6E841285CD"></a></p>
<h4>50.4 <span class="Heading">Constructors for Basic Groups</span></h4>
<p>All functions described in the previous sections call constructor operations to do the work. The names of the constructors are obtained from the names of the functions by appending <code class="code">"Cons"</code>, so for example <code class="func">CyclicGroup</code> (<a href="chap50.html#X7A7C473D87B31F3B"><span class="RefLink">50.1-2</span></a>) calls the constructor</p>
<p><code class="code">CyclicGroupCons( <var class="Arg">cat</var>, <var class="Arg">n</var> )</code></p>
<p>The first argument <var class="Arg">cat</var> for each method of this constructor must be the category for which the method is installed. For example the method for constructing a cyclic permutation group is installed as follows (see <code class="func">InstallMethod</code> (<a href="chap78.html#X837EFDAB7BEF290B"><span class="RefLink">78.2-1</span></a>) for the meaning of the arguments.</p>
<div class="example"><pre>
InstallMethod( CyclicGroupCons,
"regular perm group",
true,
[ IsPermGroup and IsRegularProp and IsFinite, IsInt and IsPosRat ], 0,
function( filter, n )
...
end );
</pre></div>
<p><a id="X82676ED5826E9E2E" name="X82676ED5826E9E2E"></a></p>
<h4>50.5 <span class="Heading">Selection Functions</span></h4>
<p><code class="code">All<var class="Arg">Library</var>Groups( <var class="Arg">fun1</var>, <var class="Arg">val1</var>, ... )</code></p>
<p>For a number of the group libraries two <em>selection functions</em> are provided. Each <code class="code">All<var class="Arg">Library</var>Groups</code> selection function permits one to select <em>all</em> groups from the library <var class="Arg">Library</var> that have a given set of properties. Currently, the library selection functions provided, of this type, are <code class="func">AllSmallGroups</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X7BB133CB7AA8F465"><span class="RefLink">smallgrp: AllSmallGroups</span></a>), <code class="func">AllIrreducibleSolvableGroups</code> (<a href="../../pkg/PrimGrp/doc/chap2.html#X7DAC64F17C8B49A2"><span class="RefLink">primgrp: AllIrreducibleSolvableGroups</span></a>), <code class="func">AllTransitiveGroups</code> (<a href="../../pkg/TransGrp/htm/CHAP001.htm#SECT002"><span class="RefLink">transgrp: AllTransitiveGroups</span></a>), and <code class="func">AllPrimitiveGroups</code> (<a href="../../pkg/PrimGrp/doc/chap1.html#X86EF380E8007D304"><span class="RefLink">primgrp: AllPrimitiveGroups</span></a>). Corresponding to each of these there is a <code class="code">One<var class="Arg">Library</var>Group</code> function (see below) which returns at most one group.</p>
<p>These functions take an arbitrary number of pairs (but at least one pair) of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values. For the function <code class="func">AllSmallGroups</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X7BB133CB7AA8F465"><span class="RefLink">smallgrp: AllSmallGroups</span></a>) the first such function must be <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), and, unlike the other library selection functions, it supports an alternative syntax where <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) is omitted (see <code class="func">AllSmallGroups</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X7BB133CB7AA8F465"><span class="RefLink">smallgrp: AllSmallGroups</span></a>)). Also, see <code class="func">AllIrreducibleSolvableGroups</code> (<a href="../../pkg/PrimGrp/doc/chap2.html#X7DAC64F17C8B49A2"><span class="RefLink">primgrp: AllIrreducibleSolvableGroups</span></a>), for details pertaining to this function.</p>
<p>For an example, let us consider the selection function for the library of transitive groups (also see <a href="../../pkg/TransGrp/htm/CHAP001.htm#SECT001"><span class="RefLink">transgrp: Transitive Permutation Groups</span></a>). The command</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllTransitiveGroups(NrMovedPoints,[10..15],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Size, [1..100],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> IsAbelian, false );</span>
</pre></div>
<p>returns a list of all transitive groups with degree between 10 and 15 and size less than 100 that are not abelian.</p>
<p>Thus <code class="code">AllTransitiveGroups</code> behaves as if it was implemented by a function similar to the one defined below, where <code class="code">TransitiveGroupsList</code> is a list of all transitive groups. (Note that in the definition below we assume for simplicity that <code class="code">AllTransitiveGroups</code> accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.)</p>
<div class="example"><pre>
AllTransitiveGroups := function( fun1, val1, fun2, val2 )
local groups, g, i;
groups := [];
for i in [ 1 .. Length( TransitiveGroupsList ) ] do
g := TransitiveGroupsList[i];
if fun1(g) = val1 or IsList(val1) and fun1(g) in val1
and fun2(g) = val2 or IsList(val2) and fun2(g) in val2
then
Add( groups, g );
fi;
od;
return groups;
end;
</pre></div>
<p>Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of properties which are precomputed for the library without having to compute them anew for each group. This will substantially speed up the selection process. In the description of each library we will list the properties that are stored for this library.</p>
<p><code class="code">One<var class="Arg">Library</var>Group( <var class="Arg">fun1</var>, <var class="Arg">val1</var>, ... )</code></p>
<p>For each <code class="code">All<var class="Arg">Library</var>Groups</code> function (see above) there is a corresponding function <code class="code">One<var class="Arg">Library</var>Group</code> on exactly the same arguments, i.e., there are <code class="func">OneSmallGroup</code> (<a href="../../pkg/SmallGrp/doc/chap1.html#X875EB1167FF6BA82"><span class="RefLink">smallgrp: OneSmallGroup</span></a>), <code class="func">OneIrreducibleSolvableGroup</code> (<a href="../../pkg/PrimGrp/doc/chap2.html#X844E60B87FC48D1B"><span class="RefLink">primgrp: OneIrreducibleSolvableGroup</span></a>), <code class="func">OneTransitiveGroup</code> (<a href="../../pkg/TransGrp/htm/CHAP001.htm#SECT002"><span class="RefLink">transgrp: OneTransitiveGroup</span></a>), and <code class="func">OnePrimitiveGroup</code> (<a href="../../pkg/PrimGrp/doc/chap1.html#X82870C177DB70470"><span class="RefLink">primgrp: OnePrimitiveGroup</span></a>). Each function simply returns <em>one</em> group in the library that has the prescribed properties, instead of <em>all</em> such groups. It returns <code class="keyw">fail</code> if no such group exists in the library.</p>
<p><a id="X7A884ECF813C2026" name="X7A884ECF813C2026"></a></p>
<h4>50.6 <span class="Heading">Finite Perfect Groups</span></h4>
<p>The <strong class="pkg">GAP</strong> library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than <span class="SimpleMath">10^6</span> excluding the following sizes:</p>
<ul>
<li><p>For <span class="SimpleMath">n = 61440</span>, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size <span class="SimpleMath">n</span> have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves.</p>
</li>
<li><p>For <span class="SimpleMath">n = 86016</span>, 368640, or 737280, the library does not yet contain the perfect groups of size <span class="SimpleMath">n</span>, it only provides their numbers which are 52, 46, and 54, respectively.</p>
</li>
</ul>
<p>Except for these eleven sizes, the list of altogether 1097 perfect groups in the library is complete. It relies on results of Derek F. Holt and Wilhelm Plesken which are published in their book "Perfect Groups" <a href="chapBib.html#biBHP89">[HP89]</a>. Moreover, they have supplied us with files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable <strong class="pkg">GAP</strong> functions whenever they are needed. Two additional groups omitted from the book "Perfect Groups" have also been included.</p>
<p>We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the <strong class="pkg">GAP</strong> community by contributing their files. It should be noted that their book contains a lot of further information for many of the library groups. So we would like to recommend it to any <strong class="pkg">GAP</strong> user who is interested in the groups.</p>
<p>The library has been brought into <strong class="pkg">GAP</strong> format by Volkmar Felsch.</p>
<p>As all groups are stored by presentations, a permutation representation is obtained by coset enumeration. Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.</p>
<p><a id="X866A25F882A4E97B" name="X866A25F882A4E97B"></a></p>
<h5>50.6-1 SizesPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizesPerfectGroups</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the ordered list of all numbers up to <span class="SimpleMath">10^6</span> that occur as sizes of perfect groups. One can iterate over the perfect groups library with:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in SizesPerfectGroups() do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for k in [1..NrPerfectLibraryGroups(n)] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> pg := PerfectGroup(n,k);</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
</pre></div>
<p><a id="X7906BBA7818E9415" name="X7906BBA7818E9415"></a></p>
<h5>50.6-2 <span class="Heading">PerfectGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">size</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectGroup</code>( [<var class="Arg">filt</var>, ]<var class="Arg">sizenumberpair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a group which is isomorphic to the library group specified by the size number <code class="code">[ <var class="Arg">size</var>, <var class="Arg">n</var> ]</code> or by the two separate arguments <var class="Arg">size</var> and <var class="Arg">n</var>, assuming a default value of <span class="SimpleMath"><var class="Arg">n</var> = 1</span>. The optional argument <var class="Arg">filt</var> defines the filter in which the group is returned. Possible filters so far are <code class="func">IsPermGroup</code> (<a href="chap43.html#X7879877482F59676"><span class="RefLink">43.1-1</span></a>) and <code class="func">IsSubgroupFpGroup</code> (<a href="chap47.html#X7AF7E2B48199452C"><span class="RefLink">47.1-1</span></a>). In the latter case, the generators and relators used coincide with those given in <a href="chapBib.html#biBHP89">[HP89]</a>. The default filter is <code class="func">IsSubgroupFpGroup</code> (<a href="chap47.html#X7AF7E2B48199452C"><span class="RefLink">47.1-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := PerfectGroup(IsPermGroup,6048,1);</span>
U3(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=PerfectGroup(IsPermGroup,823080,2);</span>
A5 2^1 19^2 C 19^1
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(G);</span>
6859
</pre></div>
<p><a id="X7E1CB2D18085FF9D" name="X7E1CB2D18085FF9D"></a></p>
<h5>50.6-3 PerfectIdentification</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PerfectIdentification</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute is set for all groups obtained from the perfect groups library and has the value <code class="code">[<var class="Arg">size</var>,<var class="Arg">nr</var>]</code> if the group is obtained with these parameters from the library.</p>
<p><a id="X7D68BE547FE5C0F5" name="X7D68BE547FE5C0F5"></a></p>
<h5>50.6-4 NumberPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberPerfectGroups</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of non-isomorphic perfect groups of size <var class="Arg">size</var> for each positive integer <var class="Arg">size</var> up to <span class="SimpleMath">10^6</span> except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns <code class="keyw">fail</code>.</p>
<p><a id="X7FE695DA86A066E1" name="X7FE695DA86A066E1"></a></p>
<h5>50.6-5 NumberPerfectLibraryGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberPerfectLibraryGroups</code>( <var class="Arg">size</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of perfect groups of size <var class="Arg">size</var> which are available in the library of finite perfect groups. (The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.)</p>
<p><a id="X866356A684F6B15E" name="X866356A684F6B15E"></a></p>
<h5>50.6-6 SizeNumbersPerfectGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizeNumbersPerfectGroups</code>( <var class="Arg">factor1</var>, <var class="Arg">factor2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SizeNumbersPerfectGroups</code> returns a list of pairs, each entry consisting of a group order and the number of those groups in the library of perfect groups that contain the specified factors <var class="Arg">factor1</var>, <var class="Arg">factor2</var>, ... among their composition factors.</p>
<p>Each argument must either be the name of a simple group or an integer which stands for the product of the sizes of one or more cyclic factors. (In fact, the function replaces all integers among the arguments by their product.)</p>
<p>The following text strings are accepted as simple group names.</p>
<ul>
<li><p><code class="code">A<var class="Arg">n</var></code> or <code class="code">A(<var class="Arg">n</var>)</code> for the alternating groups <span class="SimpleMath">A_<var class="Arg">n</var></span>, <span class="SimpleMath">5 ≤ n ≤ 9</span>, for example <code class="code">A5</code> or <code class="code">A(6)</code>.</p>
</li>
<li><p><code class="code">L<var class="Arg">n</var>(<var class="Arg">q</var>)</code> or <code class="code">L(<var class="Arg">n</var>,<var class="Arg">q</var>)</code> for PSL<span class="SimpleMath">(n,q)</span>, where <span class="SimpleMath">n ∈ { 2, 3 }</span> and <span class="SimpleMath">q</span> a prime power, ranging</p>
<ul>
<li><p>for <span class="SimpleMath">n = 2</span> from 4 to 125</p>
</li>
<li><p>for <span class="SimpleMath">n = 3</span> from 2 to 5</p>
</li>
</ul>
</li>
<li><p><code class="code">U<var class="Arg">n</var>(<var class="Arg">q</var>)</code> or <code class="code">U(<var class="Arg">n</var>,<var class="Arg">q</var>)</code> for PSU<span class="SimpleMath">(n,q)</span>, where <span class="SimpleMath">n ∈ { 3, 4 }</span> and <span class="SimpleMath">q</span> a prime power, ranging</p>
<ul>
<li><p>for <span class="SimpleMath">n = 3</span> from 3 to 5</p>
</li>
<li><p>for <span class="SimpleMath">n = 4</span> from 2 to 2</p>
</li>
</ul>
</li>
<li><p><code class="code">Sp4(4)</code> or <code class="code">S(4,4)</code> for the symplectic group Sp<span class="SimpleMath">(4,4)</span>,</p>
</li>
<li><p><code class="code">Sz(8)</code> for the Suzuki group Sz<span class="SimpleMath">(8)</span>,</p>
</li>
<li><p><code class="code">M<var class="Arg">n</var></code> or <code class="code">M(<var class="Arg">n</var>)</code> for the Mathieu groups <span class="SimpleMath">M_11</span>, <span class="SimpleMath">M_12</span>, and <span class="SimpleMath">M_22</span>, and</p>
</li>
<li><p><code class="code">J<var class="Arg">n</var></code> or <code class="code">J(<var class="Arg">n</var>)</code> for the Janko groups <span class="SimpleMath">J_1</span> and <span class="SimpleMath">J_2</span>.</p>
</li>
</ul>
<p>Note that, for most of the groups, the preceding list offers two different names in order to be consistent with the notation used in <a href="chapBib.html#biBHP89">[HP89]</a> as well as with the notation used in the <code class="func">DisplayCompositionSeries</code> (<a href="chap39.html#X82C0D0217ACB2042"><span class="RefLink">39.17-6</span></a>) command of <strong class="pkg">GAP</strong>. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like <code class="code">L2(2^5)</code> are not accepted.</p>
<p>As the use of the term PSU<span class="SimpleMath">(n,q)</span> is not unique in the literature, we mention that in this library it denotes the factor group of SU<span class="SimpleMath">(n,q)</span> by its centre, where SU<span class="SimpleMath">(n,q)</span> is the group of all <span class="SimpleMath">n × n</span> unitary matrices with entries in <span class="SimpleMath">GF(q^2)</span> and determinant 1.</p>
<p>The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.</p>
<p><a id="X845419F07BB92867" name="X845419F07BB92867"></a></p>
<h5>50.6-7 <span class="Heading">DisplayInformationPerfectGroups</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayInformationPerfectGroups</code>( <var class="Arg">size</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayInformationPerfectGroups</code>( <var class="Arg">sizenumberpair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DisplayInformationPerfectGroups</code> displays some invariants of the <var class="Arg">n</var>-th group of order <var class="Arg">size</var> from the perfect groups library.</p>
<p>If no value of <var class="Arg">n</var> has been specified, the invariants will be displayed for all groups of size <var class="Arg">size</var> available in the library.</p>
<p>Alternatively, also a list of length two may be entered as the only argument, with entries <var class="Arg">size</var> and <var class="Arg">n</var>.</p>
<p>The information provided for <span class="SimpleMath">G</span> includes the following items:</p>
<ul>
<li><p>a headline containing the size number <code class="code">[ <var class="Arg">size</var>, <var class="Arg">n</var> ]</code> of <span class="SimpleMath">G</span> in the form <code class="code"><var class="Arg">size</var>.<var class="Arg">n</var></code> (the suffix <code class="code">.<var class="Arg">n</var></code> will be suppressed if, up to isomorphism, <span class="SimpleMath">G</span> is the only perfect group of order <var class="Arg">size</var>),</p>
</li>
<li><p>a message if <span class="SimpleMath">G</span> is simple or quasisimple, i.e., if the factor group of <span class="SimpleMath">G</span> by its centre is simple,</p>
</li>
<li><p>the "description" of the structure of <span class="SimpleMath">G</span> as it is given by Holt and Plesken in <a href="chapBib.html#biBHP89">[HP89]</a> (see below),</p>
</li>
<li><p>the size of the centre of <span class="SimpleMath">G</span> (suppressed, if <span class="SimpleMath">G</span> is simple),</p>
</li>
<li><p>the prime decomposition of the size of <span class="SimpleMath">G</span>,</p>
</li>
<li><p>orbit sizes for a faithful permutation representation of <span class="SimpleMath">G</span> which is provided by the library (see below),</p>
</li>
<li><p>a reference to each occurrence of <span class="SimpleMath">G</span> in the tables of section 5.3 of <a href="chapBib.html#biBHP89">[HP89]</a>. Each of these references consists of a class number and an internal number <span class="SimpleMath">(i,j)</span> under which <span class="SimpleMath">G</span> is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( 30720, 3 );</span>
#I Perfect group 30720: A5 ( 2^4 E N 2^1 E 2^4 ) A
#I size = 2^11*3*5 orbit size = 240
#I Holt-Plesken class 1 (9,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( 30720, 6 );</span>
#I Perfect group 30720: A5 ( 2^4 x 2^4 ) C N 2^1
#I centre = 2 size = 2^11*3*5 orbit size = 384
#I Holt-Plesken class 1 (9,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );</span>
#I Perfect group 20160.1: A5 x L3(2) 2^1
#I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 5 + 16
#I Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2: A5 2^1 x L3(2)
#I centre = 2 size = 2^6*3^2*5*7 orbit sizes = 7 + 24
#I Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3: ( A5 x L3(2) ) 2^1
#I centre = 2 size = 2^6*3^2*5*7 orbit size = 192
#I Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4: simple group A8
#I size = 2^6*3^2*5*7 orbit size = 8
#I Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5: simple group L3(4)
#I size = 2^6*3^2*5*7 orbit size = 21
#I Holt-Plesken class 27 (0,1)
</pre></div>
<p><a id="X875C5BE67BAB7F71" name="X875C5BE67BAB7F71"></a></p>
<h5>50.6-8 <span class="Heading">More about the Perfect Groups Library</span></h5>
<p>For any library group <span class="SimpleMath">G</span>, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups <span class="SimpleMath">S_1, ..., S_r</span> of <span class="SimpleMath">G</span> such that there is a faithful permutation representation of <span class="SimpleMath">G</span> of degree <span class="SimpleMath">∑_{i = 1}^r [G:S_i]</span> on the set <span class="SimpleMath">{ S_i g ∣ 1 ≤ i ≤ r, g ∈ G }</span> of the cosets of the <span class="SimpleMath">S_i</span>. This allows one to construct the groups as permutation groups. The function <code class="func">DisplayInformationPerfectGroups</code> (<a href="chap50.html#X845419F07BB92867"><span class="RefLink">50.6-7</span></a>) displays only the available degree. The message</p>
<div class="example"><pre>
orbit size = 8
</pre></div>
<p>in the above example means that the available permutation representation is transitive and of degree 8, whereas the message</p>
<div class="example"><pre>
orbit sizes = 5 + 16
</pre></div>
<p>means that a nontransitive permutation representation is available which acts on two orbits of size 5 and 16 respectively.</p>
<p>The notation used in the "description" of a group is explained in section 5.1.2 of <a href="chapBib.html#biBHP89">[HP89]</a>. We quote the respective page from there:</p>
<p>Within a class <span class="SimpleMath">Q#p</span>, an isomorphism type of groups will be denoted by an ordered pair of integers <span class="SimpleMath">(r,n)</span>, where <span class="SimpleMath">r ≥ 0</span> and <span class="SimpleMath">n > 0</span>. More precisely, the isomorphism types in <span class="SimpleMath">Q # p</span> of order <span class="SimpleMath">p^r |Q|</span> will be denoted by <span class="SimpleMath">(r,1), (r,2), (r,3), ...</span>. Thus <span class="SimpleMath">Q</span> will always get the size number <span class="SimpleMath">(0,1)</span>.</p>
<p>In addition to the symbol <span class="SimpleMath">(r,n)</span>, the groups in <span class="SimpleMath">Q#p</span> will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible <span class="SimpleMath">F_pQ</span>-modules <span class="SimpleMath">M</span> with <span class="SimpleMath">|Q|.|M| ≤ 10^6</span>, where <span class="SimpleMath">F_p</span> is the field of order <span class="SimpleMath">p</span>, are assigned symbols. These will either be simply <span class="SimpleMath">p^x</span>, where <span class="SimpleMath">x</span> is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by<span class="SimpleMath">p^x</span>, <span class="SimpleMath">p^{x'}</span>, etc. The one-dimensional module with trivial <span class="SimpleMath">Q</span>-action will therefore be denoted by <span class="SimpleMath">p^1</span>. These symbols will be listed under the description of <span class="SimpleMath">Q</span>. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of <span class="SimpleMath">Q</span> itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS <a href="chapBib.html#biBCCN85">[CCN+85]</a>, for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to <span class="SimpleMath">SL(2,5)</span> by <span class="SimpleMath">A_5 2^1</span> rather than <span class="SimpleMath">2.A_5</span>.)</p>
<p>Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols.</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">×</span></strong></dt>
<dd><p>between two factors denotes a direct product of <span class="SimpleMath">F_pQ</span>-modules or groups.</p>
</dd>
<dt><strong class="Mark">C</strong></dt>
<dd><p>(for "commutator") between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form <span class="SimpleMath">(f_1 × f_2) C f_3</span> would mean that the factors <span class="SimpleMath">f_1</span> and <span class="SimpleMath">f_2</span> commute modulo <span class="SimpleMath">f_3</span> and <span class="SimpleMath">f_3</span> lies in <span class="SimpleMath">[f_1,f_2]</span>.</p>
</dd>
<dt><strong class="Mark">A</strong></dt>
<dd><p>(for "abelian") between two factors indicates that the second is in the <span class="SimpleMath">p</span>th power (but not the commutator subgroup) of the first. "A" may also follow the factors, if bracketed.</p>
</dd>
<dt><strong class="Mark">E</strong></dt>
<dd><p>(for "elementary abelian") between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting <span class="SimpleMath">F_p Q</span>-module extension does not split.</p>
</dd>
<dt><strong class="Mark">N</strong></dt>
<dd><p>(for "nonsplit") before a factor indicates that <span class="SimpleMath">Q</span> (or possibly its covering group) splits down as far at this factor but not over the factor itself. So "<span class="SimpleMath">Q f_1 N f_2</span>" means that the normal subgroup <span class="SimpleMath">f_1 f_2</span> of the group has no complement but, modulo <span class="SimpleMath">f_2</span>, <span class="SimpleMath">f_1</span>, does have a complement.</p>
</dd>
</dl>
<p>Brackets have their obvious meaning. Summarizing, we have:</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">×</span></strong></dt>
<dd><p>= direct product;</p>
</dd>
<dt><strong class="Mark">C</strong></dt>
<dd><p>= commutator subgroup;</p>
</dd>
<dt><strong class="Mark">A</strong></dt>
<dd><p>= abelian;</p>
</dd>
<dt><strong class="Mark">E</strong></dt>
<dd><p>= elementary abelian; and</p>
</dd>
<dt><strong class="Mark">N</strong></dt>
<dd><p>= nonsplit.</p>
</dd>
</dl>
<p>Here are some examples.</p>
<dl>
<dt><strong class="Mark">(i)</strong></dt>
<dd><p><span class="SimpleMath">A_5 (2^4 E 2^1 E 2^4) A</span> means that the pairs <span class="SimpleMath">2^4 E 2^1</span> and <span class="SimpleMath">2^1 E 2^4</span> are both elementary abelian of exponent 4.</p>
</dd>
<dt><strong class="Mark">(ii)</strong></dt>
<dd><p><span class="SimpleMath">A_5 (2^4 E 2^1 A) C 2^1</span> means that <span class="SimpleMath">O_2(G)</span> is of symplectic type <span class="SimpleMath">2^{1+5}</span>, with Frattini factor group of type <span class="SimpleMath">2^4 E 2^1</span>. The "A" after the <span class="SimpleMath">2^1</span> indicates that <span class="SimpleMath">G</span> has a central cyclic subgroup <span class="SimpleMath">2^1 A 2^1</span> of order 4.</p>
</dd>
<dt><strong class="Mark">(iii)</strong></dt>
<dd><p><span class="SimpleMath">L_3(2) ((2^1 E) × ( N 2^3 E 2^{3'} A) C) 2^{3'}</span> means that the <span class="SimpleMath">2^{3'}</span> factor at the bottom lies in the commutator subgroup of the pair <span class="SimpleMath">2^3 E 2^{3'}</span> in the middle, but the lower pair <span class="SimpleMath">2^{3'} A 2^{3'}</span> is abelian of exponent 4. There is also a submodule <span class="SimpleMath">2^1 E 2^{3'}</span>, and the covering group <span class="SimpleMath">L_3(2) 2^1</span> of <span class="SimpleMath">L_3(2)</span> does not split over the <span class="SimpleMath">2^3</span> factor. (Since <span class="SimpleMath">G</span> is perfect, it goes without saying that the extension <span class="SimpleMath">L_3(2) 2^1</span> cannot split itself.)</p>
</dd>
</dl>
<p>We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.</p>
<p>If such a group description has been given in the book for <span class="SimpleMath">G</span> (and, in fact, this is the case for most of the library groups), it is displayed by <code class="func">DisplayInformationPerfectGroups</code> (<a href="chap50.html#X845419F07BB92867"><span class="RefLink">50.6-7</span></a>). Otherwise the function provides a less explicit description of the (in these cases unique) Holt-Plesken class to which <span class="SimpleMath">G</span> belongs, together with a serial number if this is necessary to make it unique.</p>
<p><a id="X7873506D873EDB95" name="X7873506D873EDB95"></a></p>
<h4>50.7 <span class="Heading">Irreducible Maximal Finite Integral Matrix Groups</span></h4>
<p>A library of irreducible maximal finite integral matrix groups is provided with <strong class="pkg">GAP</strong>. It contains <span class="SimpleMath">ℚ</span>-class representatives for all of these groups of dimension at most 31, and <span class="SimpleMath">ℤ</span>-class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.</p>
<p>The groups provided in this library have been determined by Wilhelm Plesken, partially as joint work with Michael Pohst, or by members of his institute (Lehrstuhl B für Mathematik, RWTH Aachen). In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in 1979 (see <a href="chapBib.html#biBPP77">[PP77]</a>, <a href="chapBib.html#biBPP80">[PP80]</a>). The <span class="SimpleMath">ℤ</span>-class representatives of the groups of dimension 10 have been determined and computed by Bernd Souvignier (<a href="chapBib.html#biBSou94">[Sou94]</a>), and those of dimensions 11, 13, and 17 have been recomputed for this library from the circulant Gram matrices given in <a href="chapBib.html#biBPle85">[Ple85]</a>, using the stand-alone programs for the computation of short vectors and Bravais groups which have been developed in Plesken's institute. The <span class="SimpleMath">ℤ</span>-class representatives of the groups of dimensions 19 and 23 had already been determined in <a href="chapBib.html#biBPle85">[Ple85]</a>. Gabriele Nebe has recomputed them for us. Her main contribution to this library, however, is that she has determined and computed the <span class="SimpleMath">ℚ</span>-class representatives of the groups of non-prime dimensions between 12 and 24 and the groups of dimensions 25 to 31 (see <a href="chapBib.html#biBPN95">[PN95]</a>, <a href="chapBib.html#biBNP95">[NP95b]</a>, <a href="chapBib.html#biBNeb95">[Neb95]</a>, <a href="chapBib.html#biBNeb96">[Neb96]</a>).</p>
<p>The library has been brought into <strong class="pkg">GAP</strong> format by Volkmar Felsch. He has applied several <strong class="pkg">GAP</strong> routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors. We are grateful to Wilhelm Plesken, Gabriele Nebe, and Bernd Souvignier for supplying their results to <strong class="pkg">GAP</strong>.</p>
<p>In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.</p>
<p>Any integral matrix group <span class="SimpleMath">G</span> of dimension <span class="SimpleMath">n</span> is a subgroup of <span class="SimpleMath">GL_n(ℤ)</span> as well as of <span class="SimpleMath">GL_n(ℚ)</span> and hence lies in some conjugacy class of integral matrix groups under <span class="SimpleMath">GL_n(ℤ)</span> and also in some conjugacy class of rational matrix groups under <span class="SimpleMath">GL_n(ℚ)</span>. As usual, we call these classes the <span class="SimpleMath">ℤ</span>-class and the <span class="SimpleMath">ℚ</span>-class of <span class="SimpleMath">G</span>, respectively. Note that any conjugacy class of subgroups of <span class="SimpleMath">GL_n(ℚ)</span> contains at least one <span class="SimpleMath">ℤ</span>-class of subgroups of <span class="SimpleMath">GL_n(ℤ)</span> and hence can be considered as the <span class="SimpleMath">ℚ</span>-class of some integral matrix group.</p>
<p>In the context of this library we are only concerned with <span class="SimpleMath">ℤ</span>-classes and <span class="SimpleMath">ℚ</span>-classes of subgroups of <span class="SimpleMath">GL_n(ℤ)</span> which are irreducible and maximal finite in <span class="SimpleMath">GL_n(ℤ)</span> (we will call them <em>i.m.f.</em> subgroups of <span class="SimpleMath">GL_n(ℤ)</span>). We can distinguish two types of these groups:</p>
<p>First, there are those i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span> which are also maximal finite subgroups of <span class="SimpleMath">GL_n(ℚ)</span>. Let us denote the set of their <span class="SimpleMath">ℚ</span>-classes by <span class="SimpleMath">Q_1(n)</span>. It is clear from the above remark that <span class="SimpleMath">Q_1(n)</span> just consists of the <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℚ)</span>.</p>
<p>Secondly, there is the set <span class="SimpleMath">Q_2(n)</span> of the <span class="SimpleMath">ℚ</span>-classes of the remaining i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span>, i.e., of those which are not maximal finite subgroups of <span class="SimpleMath">GL_n(ℚ)</span>. For any such group <span class="SimpleMath">G</span>, say, there is at least one class <span class="SimpleMath">C ∈ Q_1(n)</span> such that <span class="SimpleMath">G</span> is conjugate under <span class="SimpleMath">ℚ</span> to a proper subgroup of some group <span class="SimpleMath">H ∈ C</span>. In fact, the class <span class="SimpleMath">C</span> is uniquely determined for any group <span class="SimpleMath">G</span> occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call <span class="SimpleMath">C</span> the <em>rational i.m.f. class</em> of <span class="SimpleMath">G</span>. Finally, we will denote the number of classes in <span class="SimpleMath">Q_1(n)</span> and <span class="SimpleMath">Q_2(n)</span> by <span class="SimpleMath">q_1(n)</span> and <span class="SimpleMath">q_2(n)</span>, respectively.</p>
<p>As an example, let us consider the case <span class="SimpleMath">n = 4</span>. There are 6 <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_4(ℤ)</span> with representative subgroups <span class="SimpleMath">G_1, ..., G_6</span> of isomorphism types <span class="SimpleMath">G_1 ≅ W(F_4)</span>, <span class="SimpleMath">G_2 ≅ D_12 ≀ C_2</span>, <span class="SimpleMath">G_3 ≅ G_4 ≅ C_2 × S_5</span>, <span class="SimpleMath">G_5 ≅ W(B_4)</span>, and <span class="SimpleMath">G_6 ≅ (D_12</span><code class="code">Y</code><span class="SimpleMath">D_12) : C_2</span>. The corresponding <span class="SimpleMath">ℚ</span>-classes, <span class="SimpleMath">R_1, ..., R_6</span>, say, are pairwise different except that <span class="SimpleMath">R_3</span> coincides with <span class="SimpleMath">R_4</span>. The groups <span class="SimpleMath">G_1</span>, <span class="SimpleMath">G_2</span>, and <span class="SimpleMath">G_3</span> are i.m.f. subgroups of <span class="SimpleMath">GL_4(ℚ)</span>, but <span class="SimpleMath">G_5</span> and <span class="SimpleMath">G_6</span> are not because they are conjugate under <span class="SimpleMath">GL_4(ℚ)</span> to proper subgroups of <span class="SimpleMath">G_1</span> and <span class="SimpleMath">G_2</span>, respectively. So we have <span class="SimpleMath">Q_1(4) = { R_1, R_2, R_3 }</span>, <span class="SimpleMath">Q_2(4) = { R_5, R_6 }</span>, <span class="SimpleMath">q_1(4) = 3</span>, and <span class="SimpleMath">q_2(4) = 2</span>.</p>
<p>The <span class="SimpleMath">q_1(n)</span> <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℚ)</span> have been determined for each dimension <span class="SimpleMath">n ≤ 31</span>. The current <strong class="pkg">GAP</strong> library provides integral representative groups for all these classes. Moreover, all <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span> are known for <span class="SimpleMath">n ≤ 11</span> and for <span class="SimpleMath">n ∈ {13,17,19,23}</span>. For these dimensions, the library offers integral representative groups for all <span class="SimpleMath">ℚ</span>-classes in <span class="SimpleMath">Q_1(n)</span> and <span class="SimpleMath">Q_2(n)</span> as well as for all <span class="SimpleMath">ℤ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_n(ℤ)</span>.</p>
<p>Any group <span class="SimpleMath">G</span> of dimension <span class="SimpleMath">n</span> given in the library is represented as the automorphism group <span class="SimpleMath">G = Aut(F,L) = { g ∈ GL_n(ℤ) ∣ Lg = L, g F g^tr = F }</span> of a positive definite symmetric <span class="SimpleMath">n × n</span> matrix <span class="SimpleMath">F ∈ ℤ^{n × n}</span> on an <span class="SimpleMath">n</span>-dimensional lattice <span class="SimpleMath">L ≅ ℤ^{1 × n}</span> (for details see e.g. <a href="chapBib.html#biBPN95">[PN95]</a>). <strong class="pkg">GAP</strong> provides for <span class="SimpleMath">G</span> a list of matrix generators and the <em>Gram matrix</em> <span class="SimpleMath">F</span>.</p>
<p>The positive definite quadratic form defined by <span class="SimpleMath">F</span> defines a <em>norm</em> <span class="SimpleMath">v F v^tr</span> for each vector <span class="SimpleMath">v ∈ L</span>, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the <em>short vectors</em>. Their set splits into orbits under <span class="SimpleMath">G</span>, and <span class="SimpleMath">G</span> being irreducible acts faithfully on each of these orbits by multiplication from the right. <strong class="pkg">GAP</strong> provides for each of these orbits the orbit size and a representative vector.</p>
<p>Like most of the other <strong class="pkg">GAP</strong> libraries, the library of i.m.f. integral matrix groups supplies an extraction function, <code class="code">ImfMatrixGroup</code>. However, as the library involves only 525 different groups, there is no need for a selection or an example function. Instead, there are two functions, <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.7-3</span></a>) and <code class="func">DisplayImfInvariants</code> (<a href="chap50.html#X8705F64B7E19DDC7"><span class="RefLink">50.7-2</span></a>), which provide some <span class="SimpleMath">ℤ</span>-class invariants that can be extracted from the library without actually constructing the representative groups themselves. The difference between these two functions is that the latter one displays the resulting data in some easily readable format, whereas the first one returns them as record components so that you can properly access them.</p>
<p>We shall give an individual description of each of the library functions, but first we would like to insert a short remark concerning their names: Any self-explaining name of a function handling <em>irreducible maximal finite integral matrix groups</em> would have to include this term in full length and hence would grow extremely long. Therefore we have decided to use the abbreviation <code class="code">Imf</code> instead in order to restrict the names to some reasonable length.</p>
<p>The first three functions can be used to formulate loops over the classes.</p>
<p><a id="X8693FD647EF3C53B" name="X8693FD647EF3C53B"></a></p>
<h5>50.7-1 ImfNumberQQClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberQQClasses</code>( <var class="Arg">dim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberQClasses</code>( <var class="Arg">dim</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfNumberZClasses</code>( <var class="Arg">dim</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">ImfNumberQQClasses</code> returns the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. rational matrix groups of dimension <var class="Arg">dim</var>. Valid values of <var class="Arg">dim</var> are all positive integers up to 31.</p>
<p>Note: In order to enable you to loop just over the classes belonging to <span class="SimpleMath">Q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span>, we have arranged the list of <span class="SimpleMath">ℚ</span>-classes of dimension <var class="Arg">dim</var> for any dimension <var class="Arg">dim</var> in the library such that, whenever the classes of <span class="SimpleMath">Q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> are known, too, i.e., in the cases <span class="SimpleMath">dim ≤ 11</span> or <span class="SimpleMath">dim ∈ {13,17,19,23}</span>, the classes of <span class="SimpleMath">Q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> precede those of <span class="SimpleMath">Q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> and hence are numbered from 1 to <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span>.</p>
<p><code class="code">ImfNumberQClasses</code> returns the number of <span class="SimpleMath">ℚ</span>-classes of groups of dimension <var class="Arg">dim</var> which are available in the library. If <span class="SimpleMath">dim ≤ 11</span> or <span class="SimpleMath">dim ∈ {13,17,19,23}</span>, this is the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">) + q_2(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_dim(ℤ)</span>. Otherwise, it is just the number <span class="SimpleMath">q_1(</span><var class="Arg">dim</var><span class="SimpleMath">)</span> of <span class="SimpleMath">ℚ</span>-classes of i.m.f. subgroups of <span class="SimpleMath">GL_dim(ℚ)</span>. Valid values of <var class="Arg">dim</var> are all positive integers up to 31.</p>
<p><code class="func">ImfNumberZClasses</code> returns the number of <span class="SimpleMath">ℤ</span>-classes in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. Valid values of <var class="Arg">dim</var> are all positive integers up to 11 and all primes up to 23.</p>
<p><a id="X8705F64B7E19DDC7" name="X8705F64B7E19DDC7"></a></p>
<h5>50.7-2 DisplayImfInvariants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayImfInvariants</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DisplayImfInvariants</code> displays the following <span class="SimpleMath">ℤ</span>-class invariants of the groups in the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>:</p>
<ul>
<li><p>its <span class="SimpleMath">ℤ</span>-class number in the form <var class="Arg">dim</var>.<var class="Arg">q</var>.<var class="Arg">z</var>, if <var class="Arg">dim</var> is at most 11 or a prime at most 23, or its <span class="SimpleMath">ℚ</span>-class number in the form <var class="Arg">dim</var>.<var class="Arg">q</var>, else,</p>
</li>
<li><p>a message if the group is solvable,</p>
</li>
<li><p>the size of the group,</p>
</li>
<li><p>the isomorphism type of the group,</p>
</li>
<li><p>the elementary divisors of the associated quadratic form,</p>
</li>
<li><p>the sizes of the orbits of short vectors (these sizes are the degrees of the faithful permutation representations which you may construct using the functions <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.7-5</span></a>) or <code class="func">IsomorphismPermGroupImfGroup</code> (<a href="chap50.html#X7CEDB6CE7BAC4518"><span class="RefLink">50.7-6</span></a>) below),</p>
</li>
<li><p>the norm of the associated short vectors,</p>
</li>
<li><p>only in case that the group is not an i.m.f. group in <span class="SimpleMath">GL_n(ℚ)</span>: an appropriate message, including the <span class="SimpleMath">ℚ</span>-class number of the corresponding rational i.m.f. class.</p>
</li>
</ul>
<p>If you specify the value 0 for any of the parameters <var class="Arg">dim</var>, <var class="Arg">q</var>, or <var class="Arg">z</var>, the command will loop over all available dimensions, <span class="SimpleMath">ℚ</span>-classes of given dimension, or <span class="SimpleMath">ℤ</span>-classes within the given <span class="SimpleMath">ℚ</span>-class, respectively. Otherwise, the values of the arguments must be in range. A value <var class="Arg">z</var> <span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 3, 1, 0 );</span>
#I Z-class 3.1.1: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = W(B3)
#I elementary divisors = 1^3
#I orbit size = 6, minimal norm = 1
#I Z-class 3.1.2: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I elementary divisors = 1*4^2
#I orbit size = 8, minimal norm = 3
#I Z-class 3.1.3: Solvable, size = 2^4*3
#I isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I elementary divisors = 1^2*4
#I orbit size = 12, minimal norm = 2
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 8, 15, 1 );</span>
#I Z-class 8.15.1: Solvable, size = 2^5*3^4
#I isomorphism type = C2 x (S3 wr S3)
#I elementary divisors = 1*3^3*9^3*27
#I orbit size = 54, minimal norm = 8
#I not maximal finite in GL(8,Q), rational imf class is 8.5
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayImfInvariants( 20, 23 );</span>
#I Q-class 20.23: Size = 2^5*3^2*5*11
#I isomorphism type = (PSL(2,11) x D12).C2
#I elementary divisors = 1^18*11^2
#I orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4
</pre></div>
<p>Note that the function <code class="func">DisplayImfInvariants</code> uses a kind of shorthand to display the elementary divisors. E. g., the expression <code class="code">1*3^3*9^3*27</code> in the preceding example stands for the elementary divisors <span class="SimpleMath">1,3,3,3,9,9,9,27</span>. (See also the next example which shows that the function <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.7-3</span></a>) provides the elementary divisors in form of an ordinary <strong class="pkg">GAP</strong> list.)</p>
<p>In the description of the isomorphism types the following notations are used:</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">x</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a direct product of a group <span class="SimpleMath">A</span> by a group <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">subd</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a subdirect product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">Y</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a central product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span> <code class="code">wr</code> <span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a wreath product of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span><code class="code">:</code><span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes a split extension of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span>,</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">A</span><code class="code">.</code><span class="SimpleMath">B</span> </strong></dt>
<dd><p>denotes just an extension of <span class="SimpleMath">A</span> by <span class="SimpleMath">B</span> (split or nonsplit).</p>
</dd>
</dl>
<p>The groups involved are</p>
<ul>
<li><p>the cyclic groups <span class="SimpleMath">C_n</span>, dihedral groups <span class="SimpleMath">D_n</span>, and generalized quaternion groups <span class="SimpleMath">Q_n</span> of order <span class="SimpleMath">n</span>, denoted by <code class="code">C</code><var class="Arg">n</var>, <code class="code">D</code><var class="Arg">n</var>, and <code class="code">Q</code><var class="Arg">n</var>, respectively,</p>
</li>
<li><p>the alternating groups <span class="SimpleMath">A_n</span> and symmetric groups <span class="SimpleMath">S_n</span> of degree <span class="SimpleMath">n</span>, denoted by <code class="code">A</code><var class="Arg">n</var> and <code class="code">S</code><var class="Arg">n</var>, respectively,</p>
</li>
<li><p>the linear groups <span class="SimpleMath">GL_n(q)</span>, <span class="SimpleMath">PGL_n(q)</span>, <span class="SimpleMath">SL_n(q)</span>, and <span class="SimpleMath">PSL_n(q)</span>, denoted by <code class="code">GL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), <code class="code">PGL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), <code class="code">SL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), and <code class="code">PSL</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the unitary groups <span class="SimpleMath">SU_n(q)</span> and <span class="SimpleMath">PSU_n(q)</span>, denoted by <code class="code">SU</code>(<var class="Arg">n</var>,<var class="Arg">q</var>) and <code class="code">PSU</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the symplectic groups <span class="SimpleMath">Sp(n,q)</span> and <span class="SimpleMath">PSp(n,q)</span>, denoted by <code class="code">Sp</code>(<var class="Arg">n</var>,<var class="Arg">q</var>) and <code class="code">PSp</code>(<var class="Arg">n</var>,<var class="Arg">q</var>), respectively,</p>
</li>
<li><p>the orthogonal groups <span class="SimpleMath">O_8^+(2)</span> and <span class="SimpleMath">PO_8^+(2)</span>, denoted by <code class="code">O+</code>(8,2) and <code class="code">PO+</code>(8,2), respectively,</p>
</li>
<li><p>the extraspecial groups <span class="SimpleMath">2_+^{1+8}</span>, <span class="SimpleMath">3_+^{1+2}</span>, <span class="SimpleMath">3_+^{1+4}</span>, and <span class="SimpleMath">5_+^{1+2}</span>, denoted by <code class="code">2+^(1+8)</code>, <code class="code">3+^(1+2)</code>, <code class="code">3+^(1+4)</code>, and <code class="code">5+^(1+2)</code>, respectively,</p>
</li>
<li><p>the Chevalley group <span class="SimpleMath">G_2(3)</span>, denoted by <code class="code">G2(3)</code>,</p>
</li>
<li><p>the twisted Chevalley group <span class="SimpleMath">^3D_4(2)</span>, denoted by <code class="code">3D4(2)</code>,</p>
</li>
<li><p>the Suzuki group <span class="SimpleMath">Sz(8)</span>, denoted by <code class="code">Sz(8)</code>,</p>
</li>
<li><p>the Weyl groups <span class="SimpleMath">W(A_n)</span>, <span class="SimpleMath">W(B_n)</span>, <span class="SimpleMath">W(D_n)</span>, <span class="SimpleMath">W(E_n)</span>, and <span class="SimpleMath">W(F_4)</span>, denoted by <code class="code">W(A<var class="Arg">n</var>)</code>, <code class="code">W(B<var class="Arg">n</var>)</code>, <code class="code">W(D<var class="Arg">n</var>)</code>, <code class="code">W(E<var class="Arg">n</var>)</code>, and <code class="code">W(F4)</code>, respectively,</p>
</li>
<li><p>the sporadic simple groups <span class="SimpleMath">Co_1</span>, <span class="SimpleMath">Co_2</span>, <span class="SimpleMath">Co_3</span>, <span class="SimpleMath">HS</span>, <span class="SimpleMath">J_2</span>, <span class="SimpleMath">M_12</span>, <span class="SimpleMath">M_22</span>, <span class="SimpleMath">M_23</span>, <span class="SimpleMath">M_24</span>, and <span class="SimpleMath">Mc</span>, denoted by <code class="code">Co1</code>, <code class="code">Co2</code>, <code class="code">Co3</code>, <code class="code">HS</code>, <code class="code">J2</code>, <code class="code">M12</code>, <code class="code">M22</code>, <code class="code">M23</code>, <code class="code">M24</code>, and <code class="code">Mc</code>, respectively,</p>
</li>
<li><p>a point stabilizer of index 11 in <span class="SimpleMath">M_11</span>, denoted by <code class="code">M10</code>.</p>
</li>
</ul>
<p>As mentioned above, the data assembled by the function <code class="func">DisplayImfInvariants</code> are "cheap data" in the sense that they can be provided by the library without loading any of its large matrix files or performing any matrix calculations. The following function allows you to get proper access to these cheap data instead of just displaying them.</p>
<p><a id="X8604A2167B2E8434" name="X8604A2167B2E8434"></a></p>
<h5>50.7-3 ImfInvariants</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfInvariants</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ImfInvariants</code> returns a record which provides some <span class="SimpleMath">ℤ</span>-class invariants of the groups in the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. A value <var class="Arg">z</var> <span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<p>The resulting record contains six or seven components:</p>
<dl>
<dt><strong class="Mark"><code class="code">size</code> </strong></dt>
<dd><p>the size of any representative group <var class="Arg">G</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">isSolvable</code> </strong></dt>
<dd><p>is <code class="keyw">true</code> if <var class="Arg">G</var> is solvable,</p>
</dd>
<dt><strong class="Mark"><code class="code">isomorphismType</code> </strong></dt>
<dd><p>a text string describing the isomorphism type of <var class="Arg">G</var> (in the same notation as used by the function <code class="code">DisplayImfInvariants</code> above),</p>
</dd>
<dt><strong class="Mark"><code class="code">elementaryDivisors</code> </strong></dt>
<dd><p>the elementary divisors of the associated Gram matrix <var class="Arg">F</var> (in the same format as the result of the function <code class="func">ElementaryDivisorsMat</code> (<a href="chap24.html#X7AC4D74F81908109"><span class="RefLink">24.9-1</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">minimalNorm</code> </strong></dt>
<dd><p>the norm of the associated short vectors,</p>
</dd>
<dt><strong class="Mark"><code class="code">sizesOrbitsShortVectors</code> </strong></dt>
<dd><p>the sizes of the orbits of short vectors under <var class="Arg">F</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">maximalQClass</code> </strong></dt>
<dd><p>the <span class="SimpleMath">ℚ</span>-class number of an i.m.f. group in <span class="SimpleMath">GL_n(ℚ)</span> that contains <var class="Arg">G</var> as a subgroup (only in case that not <var class="Arg">G</var> itself is an i.m.f. subgroup of <span class="SimpleMath">GL_n(ℚ)</span>).</p>
</dd>
</dl>
<p>Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i.m.f. class, are not only <span class="SimpleMath">ℤ</span>-class invariants, but also <span class="SimpleMath">ℚ</span>-class invariants.</p>
<p>Note further that, though the isomorphism type is a <span class="SimpleMath">ℚ</span>-class invariant, you will sometimes get different descriptions for different <span class="SimpleMath">ℤ</span>-classes of the same <span class="SimpleMath">ℚ</span>-class (as, e.g., for the classes 3.1.1 and 3.1.2 in the last example above). The purpose of this behaviour is to provide some more information about the underlying lattices.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 8, 15, 1 );</span>
rec( elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ],
isSolvable := true, isomorphismType := "C2 x (S3 wr S3)",
maximalQClass := 5, minimalNorm := 8, size := 2592,
sizesOrbitsShortVectors := [ 54 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 24, 1 ).size;</span>
10409396852733332453861621760000
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;</span>
[ 552, 53130 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [ 1 .. ImfNumberQClasses( 22 ) ] do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;</span>
C2 wr S22 = W(B22)
(C2 x PSU(6,2)).S3
(C2 x S3) wr S11 = (C2 x W(A2)) wr S11
(C2 x S12) wr C2 = (C2 x W(A11)) wr C2
C2 x S3 x S12 = C2 x W(A2) x W(A11)
(C2 x HS).C2
(C2 x Mc).C2
C2 x S23 = C2 x W(A22)
C2 x PSL(2,23)
C2 x PSL(2,23)
C2 x PGL(2,23)
C2 x PGL(2,23)
</pre></div>
<p><a id="X78935B307B909101" name="X78935B307B909101"></a></p>
<h5>50.7-4 ImfMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImfMatrixGroup</code>( <var class="Arg">dim</var>, <var class="Arg">q</var>[, <var class="Arg">z</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ImfMatrixGroup</code> is the essential extraction function of this library (note that its name has been changed from <code class="code">ImfMatGroup</code> in <strong class="pkg">GAP</strong> 3 to <code class="func">ImfMatrixGroup</code> in <strong class="pkg">GAP</strong> 4). It returns a representative group, <span class="SimpleMath">G</span> say, of the <var class="Arg">z</var>-th <span class="SimpleMath">ℤ</span>-class in the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class of i.m.f. integral matrix groups of dimension <var class="Arg">dim</var>. A value <var class="Arg">z</var><span class="SimpleMath">≠ 1</span> must not be specified if the <span class="SimpleMath">ℤ</span>-classes are not known for the given dimension, i.e., if <var class="Arg">dim</var> <span class="SimpleMath">> 11</span> and <var class="Arg">dim</var> <span class="SimpleMath">not ∈ { 13, 17, 19, 23 }</span>. The default value of <var class="Arg">z</var> is 1. This value of <var class="Arg">z</var> will be accepted even if the <span class="SimpleMath">ℤ</span>-classes are not known. Then it specifies the only representative group which is available for the <var class="Arg">q</var>-th <span class="SimpleMath">ℚ</span>-class. The greatest legal value of <var class="Arg">dim</var> is 31.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ImfMatrixGroup( 5, 1, 3 );</span>
ImfMatrixGroup(5,1,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">for m in GeneratorsOfGroup( G ) do PrintArray( m ); od;</span>
[ [ -1, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ -1, -1, -1, -1, 2 ],
[ -1, 0, 0, 0, 1 ] ]
[ [ 0, 1, 0, 0, 0 ],
[ 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1 ] ]
</pre></div>
<p>The attributes <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and <code class="code">IsSolvable</code> will be properly set in the resulting matrix group <span class="SimpleMath">G</span>. In addition, it has two attributes <code class="code">IsImfMatrixGroup</code> and <code class="code">ImfRecord</code> where the first one is just a logical flag set to <code class="keyw">true</code> and the latter one is a record. Except for the group size and the solvability flag, this record contains the same components as the resulting record of the function <code class="func">ImfInvariants</code> (<a href="chap50.html#X8604A2167B2E8434"><span class="RefLink">50.7-3</span></a>) described above, namely the components <code class="code">isomorphismType</code>, <code class="code">elementaryDivisors</code>, <code class="code">minimalNorm</code>, and <code class="code">sizesOrbitsShortVectors</code> and, if <span class="SimpleMath">G</span> is not a rational i.m.f. group, <code class="code">maximalQClass</code>. Moreover, it has the two components</p>
<dl>
<dt><strong class="Mark"><code class="code">form</code></strong></dt>
<dd><p>the associated Gram matrix <span class="SimpleMath">F</span>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">repsOrbitsShortVectors</code></strong></dt>
<dd><p>representatives of the orbits of short vectors under <span class="SimpleMath">F</span>.</p>
</dd>
</dl>
<p>The last one of these components will be required by the function <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.7-5</span></a>) below.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( G );</span>
3840
<span class="GAPprompt">gap></span> <span class="GAPinput">imf := ImfRecord( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.isomorphismType;</span>
"C2 wr S5 = C2 x W(D5)"
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintArray( imf.form );</span>
[ [ 4, 0, 0, 0, 2 ],
[ 0, 4, 0, 0, 2 ],
[ 0, 0, 4, 0, 2 ],
[ 0, 0, 0, 4, 2 ],
[ 2, 2, 2, 2, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.elementaryDivisors;</span>
[ 1, 4, 4, 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">imf.minimalNorm;</span>
4
</pre></div>
<p>If you want to perform calculations in such a matrix group <span class="SimpleMath">G</span> you should be aware of the fact that the permutation group routines of <strong class="pkg">GAP</strong> are much more efficient than the matrix group routines. Hence we recommend that you do your computations, whenever possible, in the isomorphic permutation group which is induced by the action of <span class="SimpleMath">G</span> on one of the orbits of the associated short vectors. You may call one of the following functions <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.7-5</span></a>) or <code class="func">IsomorphismPermGroupImfGroup</code> (<a href="chap50.html#X7CEDB6CE7BAC4518"><span class="RefLink">50.7-6</span></a>) to get an isomorphism to such a permutation group (note that these <strong class="pkg">GAP</strong> 4 functions have replaced the <strong class="pkg">GAP</strong> 3 functions <code class="code">PermGroup</code> and <code class="code">PermGroupImfGroup</code>).</p>
<p><a id="X84BF34B27CD5E85C" name="X84BF34B27CD5E85C"></a></p>
<h5>50.7-5 IsomorphismPermGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPermGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns an isomorphism, <span class="SimpleMath">φ</span> say, from the given i.m.f. integral matrix group <span class="SimpleMath">G</span> to a permutation group <span class="SimpleMath">P := φ(G)</span> acting on a minimal orbit, <span class="SimpleMath">S</span> say, of short vectors of <span class="SimpleMath">G</span> such that each matrix <span class="SimpleMath">m ∈ G</span> is mapped to the permutation induced by its action on <span class="SimpleMath">S</span>.</p>
<p>Note that in case of a large orbit the construction of <span class="SimpleMath">φ</span> may be space and time consuming. Fortunately, there are only six <span class="SimpleMath">ℚ</span>-classes in the library for which the smallest orbit of short vectors is of size greater than <span class="SimpleMath">20000</span>, the worst case being the orbit of size <span class="SimpleMath">196560</span> for the Leech lattice (<var class="Arg">dim</var> <span class="SimpleMath">= 24</span>, <var class="Arg">q</var> <span class="SimpleMath">= 3</span>).</p>
<p>The inverse isomorphism <span class="SimpleMath">φ^{-1}</span> from <span class="SimpleMath">P</span> to <span class="SimpleMath">G</span> is constructed by determining a <span class="SimpleMath">ℚ</span>-base <span class="SimpleMath">B ⊂ S</span> of <span class="SimpleMath">ℚ^{1 × dim}</span> in <span class="SimpleMath">S</span> and, in addition, the associated base change matrix <span class="SimpleMath">M</span> which transforms <span class="SimpleMath">B</span> into the standard base of <span class="SimpleMath">ℤ^{1 × dim}</span>. This allows a simple computation of the preimage <span class="SimpleMath">φ^{-1}(p)</span> of any permutation <span class="SimpleMath">p ∈ P</span>, as follows. If, for <span class="SimpleMath">1 ≤ i ≤</span> <var class="Arg">dim</var>, <span class="SimpleMath">b_i</span> is the position number in <span class="SimpleMath">S</span> of the <span class="SimpleMath">i</span>-th base vector in <span class="SimpleMath">B</span>, it suffices to look up the vector whose position number in <span class="SimpleMath">S</span> is the image of <span class="SimpleMath">b_i</span> under <span class="SimpleMath">p</span> and to multiply this vector by <span class="SimpleMath">M</span> to get the <span class="SimpleMath">i</span>-th row of <span class="SimpleMath">φ^{-1}(p)</span>.</p>
<p>You may use the functions <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) and <code class="func">PreImage</code> (<a href="chap32.html#X836FAEAC78B55BF4"><span class="RefLink">32.5-6</span></a>) to switch from <span class="SimpleMath">G</span> to <span class="SimpleMath">P</span> and back from <span class="SimpleMath">P</span> to <span class="SimpleMath">G</span>.</p>
<p>As an example, let us continue the preceding example and compute the solvable residuum of the group <span class="SimpleMath">G</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Perform the computations in an isomorphic permutation group.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismPermGroup( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := Image( phi );</span>
Group([ (1,7,6)(2,9)(4,5,10), (2,3,4,5)(6,9,8,7) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DerivedSubgroup( P );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( D );</span>
960
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPerfectGroup( D );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput"># We have found the solvable residuum of P,</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># now move the results back to the matrix group G.</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := PreImage( phi, D );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(R);</span>
"(C2 x C2 x C2 x C2) : A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup(D)=IdGroup(R);</span>
true
</pre></div>
<p><a id="X7CEDB6CE7BAC4518" name="X7CEDB6CE7BAC4518"></a></p>
<h5>50.7-6 IsomorphismPermGroupImfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPermGroupImfGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsomorphismPermGroupImfGroup</code> returns an isomorphism, <span class="SimpleMath">φ</span> say, from the given i.m.f. integral matrix group <var class="Arg">G</var> to a permutation group <span class="SimpleMath">P</span> acting on the <var class="Arg">n</var>-th orbit, <span class="SimpleMath">S</span> say, of short vectors of <var class="Arg">G</var> such that each matrix <span class="SimpleMath">m ∈</span> <var class="Arg">G</var> is mapped to the permutation induced by its action on <span class="SimpleMath">S</span>.</p>
<p>The only difference to the above function <code class="func">IsomorphismPermGroup</code> (<a href="chap50.html#X84BF34B27CD5E85C"><span class="RefLink">50.7-5</span></a>) is that you can specify the orbit to be used. In fact, as the orbits of short vectors are sorted by increasing sizes, the function <code class="code">IsomorphismPermGroup( <var class="Arg">G</var> )</code> has been implemented such that it is equivalent to <code class="code">IsomorphismPermGroupImfGroup( <var class="Arg">G</var>, 1 )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImfInvariants( 12, 9 ).sizesOrbitsShortVectors;</span>
[ 120, 300 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ImfMatrixGroup( 12, 9 );</span>
ImfMatrixGroup(12,9)
<span class="GAPprompt">gap></span> <span class="GAPinput">phi1 := IsomorphismPermGroupImfGroup( G, 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P1 := Image( phi1 );</span>
<permutation group of size 2400 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( P1 );</span>
120
<span class="GAPprompt">gap></span> <span class="GAPinput">phi2 := IsomorphismPermGroupImfGroup( G, 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P2 := Image( phi2 );</span>
<permutation group of size 2400 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint( P2 );</span>
300
</pre></div>
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