Database of Irreducible Matrix Groups

Magma has a database containing all irreducible subgroups of GLk(p), for p prime, k ge1 and pk < 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 pk. First are the soluble groups, followed by the insoluble. Within each subdivision, the groups are stored by increasing order. (It follows that GLk(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.

Contents

Accessing the Database

NumberOfIrreducibleMatrixGroups(k, p) : RngIntElt, RngIntElt -> RngIntElt
NumberOfSolubleIrreducibleMatrixGroups(k, p) : RngIntElt, RngIntElt -> RngIntElt
Given k and p, p prime, k ge1 and pk < 2500, NumberOfIrreducibleMatrixGroups returns the number of subgroups of GLk(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 pk < 2500, and a positive integer n, return the n-th subgroup of GLk(p) stored.

Example GrpData_IrredMat (H72E20)

We apply some of these functions to the GL4(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
V2.28, 13 July 2023