Database of Irreducible Matrix Groups

Magma has a database containing all irreducible subgroups of \(\GL_k(p)\), for \(p\) prime, \(k \ge1\), and \(p^k < 2500\). One representative of each conjugacy class of subgroups is stored.

The data used is the same as that used to store the affine primitive permutation groups. See the Primitive Groups Database section for the provenance of the data.

Within the database the groups are stored according to \(p^k\). First are the soluble groups, followed by the insoluble. Within each subdivision, the groups are stored by increasing order. (It follows that \({\operatorname{GL}}_k(p)\) is the last in each list.)

The basic access function takes three parameters, \(k\), \(p\), and number, and returns the corresponding group. Functions with name prefixed by NumberOf tell how many groups of each class there are stored.

Accessing the Database

NumberOfIrreducibleMatrixGroups(k, p): RngIntElt, RngIntElt -> RngIntElt

NumberOfSolubleIrreducibleMatrixGroups(k, p): RngIntElt, RngIntElt -> RngIntElt

Given \(k\) and \(p\), \(p\) prime, \(k \ge1\), and \(p^k < 2500\), NumberOfIrreducibleMatrixGroups returns the number of subgroups of \(\GL_k(p)\) stored. The other function returns the number of soluble subgroups stored.

IrreducibleMatrixGroup(k, p, n): RngIntElt, RngIntElt, RngIntElt -> GrpMat

Given \(k\) and \(p\) \(p\) prime, \(k \ge1\), and \(p^k < 2500\), and a positive integer \(n\), return the \(n\)-th subgroup of \(\GL_k(p)\) stored.

Example: Irreducible Matrix Groups in GL(4, 5) (ex-462c7c)

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We apply some of these functions to the \({\operatorname{GL}}_4(5)\) case.

> NumberOfIrreducibleMatrixGroups(4, 5);
647
> NumberOfSolubleIrreducibleMatrixGroups(4, 5);
509
> G := IrreducibleMatrixGroup(4, 5, 511);
> ChiefFactors(G);
    G
    |  Cyclic(2)
    *
    |  Alternating(5)
    *
    |  Cyclic(2)
    1
> IsIrreducible(G);
true
> IsAbsolutelyIrreducible(G);
false

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