The package IRREDSOL provides a library of irreducible soluble subgroups of matrix groups over finite fields and a corresponding library of primitive soluble groups.

Currently, IRREDSOL contains all subgroups, up to conjugacy, of `GL(n, q)`,
where `n` is a positive integer and `q`
is a prime power satisfying `q ^{n} leq2^{24} - 1 = 16,777,215`. The underlying data base consists of

The groups in the IRREDSOL
library can be accessed one at a time (see Section Low level access functions). In addition, there are functions which allow to
search the library for groups with given properties (see Section Finding matrix groups with given properties). Moreover, given an irreducible soluble matrix group
`G`, it is possible to identify the group in the library to which `G` is conjugate,
including a conjugating matrix, if desired. See Section Identification of irreducible groups.

Apart from this, the IRREDSOL package provides additional functionality for matrix groups, such as the computation of imprimitivity systems; see Chapter Additional functionality for matrix groups.

It is well-known that there is a bijection between the irreducible soluble subgroups of
`GL(n, p)`, where
`p` is a prime, and the conjugacy classes, or equivalently the isomorphism types, of
primitive soluble subgroups of `Sym(p ^{n})`. The IRREDSOL package contains
functions to translate between irreducible soluble matrix groups and primitive
groups, to search for primitive soluble groups with given properties, and functions to
recognise them, up to isomorphism (or, equivalently, up to conjugacy in

Note that GAP contains another library consisting of all `372` irreducible soluble
subgroups of `GL(n, p)`, where `n > 1`, `p` is a prime, and `p ^{n} < 2^{8}`. This library
was originally
created by Mark Short Sho, and two omissions in

The groups in the IRREDSOL data base were constructed using the Aschbacher
classification Asc of maximal subgroups of linear groups. Further details can be found
in EH, where the
construction of all irreducible soluble subgroups of `GL(n, q)` with `q ^{n} < 3^{8}`
is described.

For a historical account of the classification of irreducible matrix groups and primitive permutation groups, the reader is referred to Sho and, for more recent developments, to EH.

IRREDSOL manual

November 2022