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2 Irreducible Matrix Groups
2.1 Irreducible Solvable Matrix Groups
2.1-1 IrreducibleSolvableGroupMS
2.1-2 NumberIrreducibleSolvableGroups
2.1-3 AllIrreducibleSolvableGroups
2.1-4 OneIrreducibleSolvableGroup
2.1-5 PrimitiveIndexIrreducibleSolvableGroup
2.1-6 IrreducibleSolvableGroup

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

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 \(= 13\).

##### 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 \(f(G) \in v\), 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`>.

##### 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 \(f(G) \in v\), 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`>.

##### 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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