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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 is constructed by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [OVL05], [DE05], contains the following groups:

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

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

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The groups of order dividing p6 for p a prime.

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The groups of order qn p, where qn is a prime-power dividing 28, 36, 55 or 74 and p is a prime different to q.

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The groups of square-free order. For a different mechanism for accessing the p-groups in this collection, see Section The p-groups of Order Dividing p7, specifically the functions SearchPGroups and CountPGroups. These functions also access groups of order p7.

p-groups: Magma contains the means to construct all p-groups of order pn where n≤7. The data used in the constructions was supplied by Hans Ulrich Besche, Bettina Eick, Eamonn O'Brien, Mike Newman and Michael Vaughan-Lee [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [OVL05].

Metacyclic p-groups: Magma is able to construct all metacyclic groups of order pn. This machinery was developed by Mike Newman, Eamonn O'Brien, and Michael Vaughan-Lee.

Perfect Groups: This database contains all perfect groups up to order 50000, and many classes of perfect groups up to order one million. Each group is defined by means of a finite presentation. Further information is also provided which allows the construction of permutation representations. This database was constructed by Derek Holt and Willem Plesken [HP89].

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 S is one of M24, HS, J3, McL, Sz(32) or L6(2).

Transitive Permutation Groups: This database is a Magma version of the database of transitive permutation groups constructed by Alexander Hulpke [Hul05] (for degree up to 30) and Cannon and Holt [CH08]. It contains all transitive permutation groups having degree up to 32.

Primitive Permutation Groups: This is a database containing all primitive permutation groups having degree less than 4095 as determined by Sims (for degree ≤50), Roney-Dougal and Unger [RDU03] (for degree < 1000), Roney-Dougal [RD05] (for degree < 2500), and Coutts, Quick and Roney-Dougal [CQRD11] (for degree < 4096).

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 GLn(( 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].

Irreducible Matrix Groups: A database of the irreducible subgroups of GLn(p), p prime, n ≥1 and pn < 2500. The groups were determined by Colva Roney-Dougal and William Unger [RDU03] (for pn < 1000) and Roney-Dougal [RD05].

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

ATLAS Groups: This database contains representations of nearly simple groups, as in the Birmingham ATLAS of Finite Group Representations. The data was supplied by Rob Wilson.

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013