Introduction

This chapter describes the use of the various databases of groups that form part of Magma. The available databases are as follows:

Small Groups: This database, constructed by Hans Ulrich Beske, Bettina Eick and Eamonn O'Brien [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], contains the following groups.

-

All groups of order up to 2000, excluding the groups of order 1024.

-

The groups of orders 5^5 and 7^4.

-

The groups whose order is a product of at most 3 distinct primes.

-

The groups of order q^n p, where q^n is a prime-power dividing 2^8, 3^6, 5^5 or 7^4 and p is a prime different to q.

Almost Simple Groups: This database contains information about every group G, where S <= G <= Aut(S) and S is a simple group of order less than 16000000 or M_(24).

Transitive Permutation Groups: This database is a Magma version of the database of transitive permutation groups constructed by Alexander Hulpke [Hul]. It contains all transitive permutation groups having degree up to 30.

Primitive Permutation Groups: This is a database containing all primitive permutation groups having degree less than 1000 as determined by Colva Roney-Dougal and William Unger [RDU03].

Rational Maximal Matrix Groups: This contains the rational maximal finite matrix groups and their invariant forms, for small dimensions (up to 31) as determined by Gabi Nebe and Willem Plesken [NP95], [Neb96]. Each entry can be accessed either as a matrix group or as a lattice.

Quaternionic Matrix Groups: A database of the finite absolutely irreducible subgroups of GL_n(( D)) where ( D) is a definite quaternion algebra whose centre has degree d over Q and nd leq10. Each entry can be accessed either as a matrix group or as a lattice. The database was constructed by Gabi Nebe [Neb98].

Soluble Primitive Groups: This database contains one representative of each conjugacy class of irreducible soluble subgroups of ( GL)(n, p), p prime, for n > 1 and p^n < 256. It was constructed by Michael Short [Sho92].