/usr/share/gap/pkg/PrimGrp/doc/chap2.txt

  
  2 Irreducible Matrix Groups
  
  
  2.1 Irreducible Solvable Matrix Groups
  
  2.1-1 IrreducibleSolvableGroupMS
  
  IrreducibleSolvableGroupMS( n, p, i )  function
  
  This  function  returns  a  representative  of  the  i-th conjugacy class of
  irreducible solvable subgroup of GL(n, p), where n is an integer > 1, p is a
  prime, and p^n < 256.
  
  The  numbering  of  the  representatives  should  be  considered  arbitrary.
  However,  it  is guaranteed that the i-th group on this list will lie in the
  same  conjugacy  class  in  all future versions of GAP, unless two (or more)
  groups  on  the  list  are  discovered  to  be  duplicates,  in  which  case
  IrreducibleSolvableGroupMS   will  return  fail  for  all  but  one  of  the
  duplicates.
  
  For  values  of  n, p, and i admissible to IrreducibleSolvableGroup (2.1-6),
  IrreducibleSolvableGroupMS  returns  a  representative of the same conjugacy
  class  of  subgroups  of  GL(n, p) as IrreducibleSolvableGroup (2.1-6). Note
  that  it  currently  adds two more groups (missing from the original list by
  Mark Short) for n = 2, p = 13.
  
  2.1-2 NumberIrreducibleSolvableGroups
  
  NumberIrreducibleSolvableGroups( n, p )  function
  
  This  function  returns  the  number  of  conjugacy  classes  of irreducible
  solvable subgroup of GL(n, p).
  
  2.1-3 AllIrreducibleSolvableGroups
  
  AllIrreducibleSolvableGroups( func1, val1, func2, val2, ... )  function
  
  This  function returns a list of conjugacy class representatives G of matrix
  groups  over  a  prime  field  such that f(G) = v or f(G) ∈ v, for all pairs
  (f,v)  in (func1, val1), (func2, val2), .... The following possibilities for
  the  functions  f are particularly efficient, because the values can be read
  off  the  information  in  the  data base: DegreeOfMatrixGroup (or Dimension
  (Reference:     Dimension)     or     DimensionOfMatrixGroup     (Reference:
  DimensionOfMatrixGroup))  for  the linear degree, Characteristic (Reference:
  Characteristic)  for  the  field  characteristic,  Size  (Reference:  Size),
  IsPrimitiveMatrixGroup (or IsLinearlyPrimitive), and MinimalBlockDimension>.
  
  2.1-4 OneIrreducibleSolvableGroup
  
  OneIrreducibleSolvableGroup( func1, val1, func2, val2, ... )  function
  
  This function returns one solvable subgroup G of a matrix group over a prime
  field  such that f(G) = v or f(G) ∈ v, for all pairs (f,v) in (func1, val1),
  (func2,  val2),  ....  The  following  possibilities for the functions f are
  particularly  efficient,  because the values can be read off the information
  in  the  data base: DegreeOfMatrixGroup (or Dimension (Reference: Dimension)
  or   DimensionOfMatrixGroup  (Reference:  DimensionOfMatrixGroup))  for  the
  linear  degree,  Characteristic  (Reference:  Characteristic)  for the field
  characteristic,   Size   (Reference:   Size),   IsPrimitiveMatrixGroup   (or
  IsLinearlyPrimitive), and MinimalBlockDimension>.
  
  2.1-5 PrimitiveIndexIrreducibleSolvableGroup
  
  PrimitiveIndexIrreducibleSolvableGroup  global variable
  
  This  variable  provides  a  way  to get from irreducible solvable groups to
  primitive groups and vice versa. For the group G = IrreducibleSolvableGroup(
  n,      p,      k      )      and      d      =      p^n,      the     entry
  PrimitiveIndexIrreducibleSolvableGroup[d][i]  gives  the index number of the
  semidirect product p^n:G in the library of primitive groups.
  
  Searching  for  an  index  in  this list with Position (Reference: Position)
  gives the translation in the other direction.
  
  2.1-6 IrreducibleSolvableGroup
  
  IrreducibleSolvableGroup( n, p, i )  function
  
  This  function  is  obsolete,  because  for  n  = 2, p = 13, two groups were
  missing  from  the underlying database. It has been replaced by the function
  IrreducibleSolvableGroupMS  (2.1-1).  Please  note  that the latter function
  does  not guarantee any ordering of the groups in the database. However, for
  values   of   n,   p,   and   i   admissible   to  IrreducibleSolvableGroup,
  IrreducibleSolvableGroupMS  (2.1-1)  returns  a  representative  of the same
  conjugacy  class  of  subgroups  of GL(n, p) as IrreducibleSolvableGroup did
  before.
  

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