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  1 Primitive Permutation Groups
  
  
  1.1 Primitive Permutation Groups
  
  GAP contains a library of primitive permutation groups which includes, up to
  permutation   isomorphism  (i.e.,  up  to  conjugacy  in  the  corresponding
  symmetric  group),  all  primitive  permutation  groups  of  degree  < 4096,
  calculated in [RD05] and [Qui11], in particular,
  
      the  primitive  permutation  groups  up  to  degree 50,  calculated by
        C. Sims,
  
      the  primitive  groups  with  insoluble  socles  of  degree  < 1000 as
        calculated in [DM88],
  
      the  solvable  (hence  affine)  primitive permutation groups of degree
        < 256 as calculated by M. Short [Sho92],
  
      some insolvable affine primitive permutation groups of degree < 256 as
        calculated in [The97].
  
      The  solvable  primitive  groups  of degree up to 999 as calculated in
        [EH03].
  
      The  primitive groups of affine type of degree up to 999 as calculated
        in [RDU03].
  
  Not  all groups are named, those which do have names use ATLAS notation. Not
  all names are necessary unique!
  
  The  list  given  in  [RD05]  is believed to be complete, correcting various
  omissions in [DM88], [Sho92] and [The97].
  
  In  detail,  we  guarantee  the  following  properties  for this and further
  versions (but not versions which came before GAP 4.2) of the library:
  
      All  groups  in  the  library  are primitive permutation groups of the
        indicated degree.
  
      The  positions  of  the  groups  in  the  library  are stable. That is
        PrimitiveGroup(n,nr)  will  always  give  you a permutation isomorphic
        group.  Note  however  that  we  do  not  guarantee to keep the chosen
        S_n-representative, the generating set or the name for eternity.
  
      Different groups in the library are not conjugate in S_n.
  
      If  a  group  in  the  library  has a primitive subgroup with the same
        socle, this group is in the library as well.
  
  (Note  that  the arrangement of groups is not guaranteed to be in increasing
  size, though it holds for many degrees.)
  
  The  selection  functions  (see 'Reference:  Selection  Functions')  for the
  primitive  groups library are AllPrimitiveGroups and OnePrimitiveGroup. They
  obtain  the following properties from the database without having to compute
  them anew:
  
  NrMovedPoints   (Reference:  NrMovedPoints  for  a  list  or  collection  of
  permutations), Size (Reference: Size), Transitivity (Reference: Transitivity
  for a group and an action domain), ONanScottType (Reference: ONanScottType),
  IsSimpleGroup   (Reference:   IsSimpleGroup),   IsSolvableGroup  (Reference:
  IsSolvableGroup),        and       SocleTypePrimitiveGroup       (Reference:
  SocleTypePrimitiveGroup).
  
  (Note,  that for groups of degree up to 2499, O'Nan-Scott types 4a, 4b and 5
  cannot occur.)
  
  1.1-1 PrimitiveGroup
  
  PrimitiveGroup( deg, nr )  function
  
  returns  the  primitive  permutation group of degree deg with number nr from
  the list.
  
  The  arrangement  of  the groups of degrees not greater than 50 differs from
  the  arrangement  of primitive groups in the list of C. Sims, which was used
  in GAP 3. See SimsNo (1.2-2).
  
  1.1-2 NrPrimitiveGroups
  
  NrPrimitiveGroups( deg )  function
  
  returns  the  number  of  primitive  permutation groups of degree deg in the
  library.
  
    Example  
    gap> NrPrimitiveGroups(25);
    28
    gap> PrimitiveGroup(25,19);
    5^2:((Q(8):3)'4)
    gap> PrimitiveGroup(25,20);
    ASL(2, 5)
    gap> PrimitiveGroup(25,22);
    AGL(2, 5)
    gap> PrimitiveGroup(25,23);
    (A(5) x A(5)):2
  
  
  1.1-3 AllPrimitiveGroups
  
  AllPrimitiveGroups( attr1, val1, attr2, val2, ... )  function
  
  This  is  a  selection  function which permits to select all groups from the
  Primitive  Group  Library  that  have  a given set of properties. It accepts
  arguments  as  specified  in Section 'Reference: Selection Functions' of the
  GAP reference manual.
  
  1.1-4 OnePrimitiveGroup
  
  OnePrimitiveGroup( attr1, val1, attr2, val2, ... )  function
  
  This  is a selection function which permits to select at most one group from
  the  Primitive Group Library that have a given set of properties. It accepts
  arguments  as  specified  in Section 'Reference: Selection Functions' of the
  GAP reference manual.
  
  1.1-5 PrimitiveGroupsIterator
  
  PrimitiveGroupsIterator( attr1, val1, attr2, val2, ... )  function
  
  returns  an  iterator  through AllPrimitiveGroups(attr1,val1,attr2,val2,...)
  without creating all these groups at the same time.
  
  1.1-6 COHORTS_PRIMITIVE_GROUPS
  
  COHORTS_PRIMITIVE_GROUPS global variable
  
  In  [DM88]  the  primitive  groups  are sorted in cohorts according to their
  socle. For each degree less than 2500, the variable COHORTS_PRIMITIVE_GROUPS
  contains a list of the cohorts for the primitive groups of this degree. Each
  cohort  is  represented by a list of length 2, the first entry specifies the
  socle        type       (see       SocleTypePrimitiveGroup       (Reference:
  SocleTypePrimitiveGroup)), the second entry listing the index numbers of the
  groups in this degree.
  
  For  example  in  degree  49,  we have four cohorts with socles (ℤ / 7 ℤ)^2,
  L_2(7)^2,  A_7^2  and  A_49  respectively.  the group PrimitiveGroup(49,36),
  which  is isomorphic to (A_7 × A_7):2^2, lies in the third cohort with socle
  (A_7 × A_7).
  
    Example  
    gap> COHORTS_PRIMITIVE_GROUPS[49];
    [ [ rec( parameter := 7, series := "Z", width := 2 ), 
          [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 
              20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 ] ], 
      [ rec( parameter := [ 2, 7 ], series := "L", width := 2 ), [ 34 ] ], 
      [ rec( parameter := 7, series := "A", width := 2 ), [ 35, 36, 37, 38 ] ], 
      [ rec( parameter := 49, series := "A", width := 1 ), [ 39, 40 ] ] ]
  
  
  
  1.2 Index numbers of primitive groups
  
  1.2-1 PrimitiveIdentification
  
  PrimitiveIdentification( G )  attribute
  
  For  a  primitive permutation group for which an S_n-conjugate exists in the
  library  of  primitive  permutation groups (see 1.1), this attribute returns
  the     index     position.     That     is     G     is     conjugate    to
  PrimitiveGroup(NrMovedPoints(G),PrimitiveIdentification(G)).
  
  Methods only exist if the primitive groups library is installed.
  
  Note: As this function uses the primitive groups library, the result is only
  guaranteed  to  the  same  extent  as  this  library.  If  it is incomplete,
  PrimitiveIdentification  might  return  an existing index number for a group
  not in the library.
  
    Example  
    gap> PrimitiveIdentification(Group((1,2),(1,2,3)));
    2
  
  
  1.2-2 SimsNo
  
  SimsNo( G )  attribute
  
  If  G  is  a  primitive  group  of  degree  not greater than 50, obtained by
  PrimitiveGroup  (1.1-1)  (respectively one of the selection functions), then
  this  attribute  contains the number of the isomorphic group in the original
  list of C. Sims. (This is the arrangement as it was used in GAP 3.)
  
    Example  
    gap> g:=PrimitiveGroup(25,2);
    5^2:S(3)
    gap> SimsNo(g);
    3
  
  
  As  mentioned in the previous section, the index numbers of primitive groups
  in  GAP are guaranteed to remain stable. (Thus, missing groups will be added
  to  the  library  at  the  end of each degree.) In particular, it is safe to
  refer to a primitive group of type deg, nr in the GAP library.
  
  1.2-3 PRIMITIVE_INDICES_MAGMA
  
  PRIMITIVE_INDICES_MAGMA global variable
  
  The system Magma also provides a list of primitive groups (see [RDU03]). For
  historical  reasons, its indexing up to degree 999 differs from the one used
  by  GAP.  The  variable  PRIMITIVE_INDICES_MAGMA  can be used to obtain this
  correspondence.    The    magma    index    number    of   the   GAP   group
  PrimitiveGroup(deg,nr)        is       stored       in       the       entry
  PRIMITIVE_INDICES_MAGMA[deg][nr], for degree at most 999.
  
  Vice  versa,  the group of degree deg with Magma index number nr has the GAP
  index
  
  Position(PRIMITIVE_INDICES_MAGMA[deg],nr),  in particular it can be obtained
  by the GAP command
  
  PrimitiveGroup(deg,Position(PRIMITIVE_INDICES_MAGMA[deg],nr));