`‣ 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 \(\textit{p}^{\textit{n}} < 256\).

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` \(= 13\).

`‣ NumberIrreducibleSolvableGroups` ( n, p ) | ( function ) |

This function returns the number of conjugacy classes of irreducible solvable subgroup of GL(`n`, `p`).

`‣ 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 \(f(G) \in v\), for all pairs \((f,v)\) in (`func1`, `val1`), (`func2`, `val2`), \(\ldots\). 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`

>.

`‣ 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 \(f(G) \in v\), for all pairs \((f,v)\) in (`func1`, `val1`), (`func2`, `val2`), \(\ldots\). 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`

>.

`‣ 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( `

and \(d = p^n\), the entry `n`, `p`, `k` )`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.

`‣ 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.

generated by GAPDoc2HTML