This chapter describes the use of the various databases of groups that form part of Magma. The available databases are as follows:
Simple Groups: This database contains all the simple groups of order less than 1020. While Magma contains tools that allow the user to construct any finite simple group the purpose of this facility is to make it easier to access a simple group and to run through the simple groups in order of increasing group order.
Small Groups: This database is constructed by Hans Ulrich Besche, Heiko Dietrich, Bettina Eick, Eamonn O'Brien, and Eileen Pan [BE99a], [BEO01], [BE99b], [O'B90], [BE01], [O'B91], [MNVL04], [OVL05], [DE05], [DEP22], and contains the following groups:
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 A. Hulpke [Hul05] (for degree up to 30), J. Cannon and D. Holt [CH08] (degree 32), D. Holt and G. Royle [HR19] (degrees 33 to 47), and D. Holt (degree 48). It contains all transitive permutation groups having degree up to 48.
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.
Fundamental Groups of 3-Manifolds: This database consists of the fundamental groups of the 10,986 small-volume closed hyperbolic manifolds in the Hodgson--Weeks census.
Automatic Groups of 3-Manifolds: This database contains automatic groups for 5,389 of the 10,986 small-volume closed hyperbolic manifolds in the Hodgson--Weeks census.