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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:
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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 4 primes.
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- The groups of order dividing p7 for p a prime.
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- The groups of order 38.
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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
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 up to 8191 as determined by Sims (for degree ≤50),
Roney-Dougal and Unger [RDU03] (for degree < 1000),
Roney-Dougal [RD05] (for degree < 2500),
Coutts, Quick, and Roney-Dougal [CQRD11] (for degree < 4096),
and Stratford (for degree < 8192).
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.
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