  
  [1X2 [33X[0;0YIrreducible Matrix Groups[133X[101X
  
  
  [1X2.1 [33X[0;0YIrreducible Solvable Matrix Groups[133X[101X
  
  [1X2.1-1 IrreducibleSolvableGroupMS[101X
  
  [33X[1;0Y[29X[2XIrreducibleSolvableGroupMS[102X( [3Xn[103X, [3Xp[103X, [3Xi[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns  a  representative  of  the  [3Xi[103X-th conjugacy class of
  irreducible solvable subgroup of GL([3Xn[103X, [3Xp[103X), where [3Xn[103X is an integer [22X> 1[122X, [3Xp[103X is a
  prime, and [22X[3Xp[103X^[3Xn[103X < 256[122X.[133X
  
  [33X[0;0YThe  numbering  of  the  representatives  should  be  considered  arbitrary.
  However,  it  is guaranteed that the [3Xi[103X-th group on this list will lie in the
  same  conjugacy  class  in  all future versions of [5XGAP[105X, unless two (or more)
  groups  on  the  list  are  discovered  to  be  duplicates,  in  which  case
  [2XIrreducibleSolvableGroupMS[102X   will  return  [9Xfail[109X  for  all  but  one  of  the
  duplicates.[133X
  
  [33X[0;0YFor  values  of  [3Xn[103X, [3Xp[103X, and [3Xi[103X admissible to [2XIrreducibleSolvableGroup[102X ([14X2.1-6[114X),
  [2XIrreducibleSolvableGroupMS[102X  returns  a  representative of the same conjugacy
  class  of  subgroups  of  GL([3Xn[103X, [3Xp[103X) as [2XIrreducibleSolvableGroup[102X ([14X2.1-6[114X). Note
  that  it  currently  adds two more groups (missing from the original list by
  Mark Short) for [3Xn[103X [22X= 2[122X, [3Xp[103X [22X= 13[122X.[133X
  
  [1X2.1-2 NumberIrreducibleSolvableGroups[101X
  
  [33X[1;0Y[29X[2XNumberIrreducibleSolvableGroups[102X( [3Xn[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns  the  number  of  conjugacy  classes  of irreducible
  solvable subgroup of GL([3Xn[103X, [3Xp[103X).[133X
  
  [1X2.1-3 AllIrreducibleSolvableGroups[101X
  
  [33X[1;0Y[29X[2XAllIrreducibleSolvableGroups[102X( [3Xfunc1[103X, [3Xval1[103X, [3Xfunc2[103X, [3Xval2[103X, [3X...[103X ) [32X function[133X
  
  [33X[0;0YThis  function returns a list of conjugacy class representatives [22XG[122X of matrix
  groups  over  a  prime  field  such that [22Xf(G) = v[122X or [22Xf(G) ∈ v[122X, for all pairs
  [22X(f,v)[122X  in ([3Xfunc1[103X, [3Xval1[103X), ([3Xfunc2[103X, [3Xval2[103X), [22X...[122X. The following possibilities for
  the  functions  [22Xf[122X are particularly efficient, because the values can be read
  off  the  information  in  the  data base: [10XDegreeOfMatrixGroup[110X (or [2XDimension[102X
  ([14XReference:     Dimension[114X)     or     [2XDimensionOfMatrixGroup[102X     ([14XReference:
  DimensionOfMatrixGroup[114X))  for  the linear degree, [2XCharacteristic[102X ([14XReference:
  Characteristic[114X)  for  the  field  characteristic,  [2XSize[102X  ([14XReference:  Size[114X),
  [10XIsPrimitiveMatrixGroup[110X (or [10XIsLinearlyPrimitive[110X), and [10XMinimalBlockDimension[110X>.[133X
  
  [1X2.1-4 OneIrreducibleSolvableGroup[101X
  
  [33X[1;0Y[29X[2XOneIrreducibleSolvableGroup[102X( [3Xfunc1[103X, [3Xval1[103X, [3Xfunc2[103X, [3Xval2[103X, [3X...[103X ) [32X function[133X
  
  [33X[0;0YThis function returns one solvable subgroup [22XG[122X of a matrix group over a prime
  field  such that [22Xf(G) = v[122X or [22Xf(G) ∈ v[122X, for all pairs [22X(f,v)[122X in ([3Xfunc1[103X, [3Xval1[103X),
  ([3Xfunc2[103X,  [3Xval2[103X),  [22X...[122X.  The  following  possibilities for the functions [22Xf[122X are
  particularly  efficient,  because the values can be read off the information
  in  the  data base: [10XDegreeOfMatrixGroup[110X (or [2XDimension[102X ([14XReference: Dimension[114X)
  or   [2XDimensionOfMatrixGroup[102X  ([14XReference:  DimensionOfMatrixGroup[114X))  for  the
  linear  degree,  [2XCharacteristic[102X  ([14XReference:  Characteristic[114X)  for the field
  characteristic,   [2XSize[102X   ([14XReference:   Size[114X),   [10XIsPrimitiveMatrixGroup[110X   (or
  [10XIsLinearlyPrimitive[110X), and [10XMinimalBlockDimension[110X>.[133X
  
  [1X2.1-5 PrimitiveIndexIrreducibleSolvableGroup[101X
  
  [33X[1;0Y[29X[2XPrimitiveIndexIrreducibleSolvableGroup[102X[32X global variable[133X
  
  [33X[0;0YThis  variable  provides  a  way  to get from irreducible solvable groups to
  primitive groups and vice versa. For the group [22XG[122X = [10XIrreducibleSolvableGroup(
  [3Xn[103X[10X,      [3Xp[103X[10X,      [3Xk[103X[10X      )[110X      and      [22Xd      =      p^n[122X,      the     entry
  [10XPrimitiveIndexIrreducibleSolvableGroup[d][i][110X  gives  the index number of the
  semidirect product [22Xp^n:G[122X in the library of primitive groups.[133X
  
  [33X[0;0YSearching  for  an  index  in  this list with [2XPosition[102X ([14XReference: Position[114X)
  gives the translation in the other direction.[133X
  
  [1X2.1-6 IrreducibleSolvableGroup[101X
  
  [33X[1;0Y[29X[2XIrreducibleSolvableGroup[102X( [3Xn[103X, [3Xp[103X, [3Xi[103X ) [32X function[133X
  
  [33X[0;0YThis  function  is  obsolete,  because  for  [3Xn[103X  [22X= 2[122X, [3Xp[103X [22X= 13[122X, two groups were
  missing  from  the underlying database. It has been replaced by the function
  [2XIrreducibleSolvableGroupMS[102X  ([14X2.1-1[114X).  Please  note  that the latter function
  does  not guarantee any ordering of the groups in the database. However, for
  values   of   [3Xn[103X,   [3Xp[103X,   and   [3Xi[103X   admissible   to  [2XIrreducibleSolvableGroup[102X,
  [2XIrreducibleSolvableGroupMS[102X  ([14X2.1-1[114X)  returns  a  representative  of the same
  conjugacy  class  of  subgroups  of GL([3Xn[103X, [3Xp[103X) as [2XIrreducibleSolvableGroup[102X did
  before.[133X
  
