The Small Groups library

The Small Groups library provides access to descriptions of the groups of ``small" order. The groups are listed up to isomorphism. At present, the library contains the following groups:

those of order at most 2000 except 1024 (423 164 062 groups);
those of cubefree order at most 50000 (395 703 groups);
those of order p^7 for p = 3,5,7,11 (907 489 groups);
those of order p^n for n at most 6 and p prime;
those of order q^n * p for q^n dividing 2^8, 3^6, 5^5 or 7^4 and p a prime different to q;
those of squarefree order;
those whose order factorises into at most 3 primes.

This library has been prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien. The groups have been been constructed by the authors of the library with the help of Mike Newman (order p^4, p^5 and p^6), Boris Girnat (order p^5),Mike Vaughan-Lee (orders p^6 and p^7) and Heiko Dietrich (squarefree and cubefree orders).

A history of group constructions

The construction of this library

The number of groups of order at most 2000

References to the construction of the Small Groups library

How to obtain and use the Small Groups library

An extended report on the technical details on the Small Groups library

History of group constructions

The construction of all groups of a given order is one of the oldest problems in finite group theory. It was initiated by Cayley in 1854 when he determined all groups of order 4 and 6.

Historically, the approaches to this problem involved a large number of hand-computations and case distinctions, and focused on very specific properties of the groups. They were consequently ad-hoc in nature, and many contained significant errors.

For the determination of the Small Groups library, we developed practical algorithms to construct the groups of a given order. While these methods rely on group-theoretic properties, they are inherently general-purpose and their results appear reliable.

A survey of the history of group construction is available in ``A millennium project: constructing Small Groups'', Internat. J. Algebra Comput. 12, 623 - 644 (2002) by Besche, Eick and O'Brien.

The construction of the Small Groups library

The library was developed in various stages. It incorporates the 2-and 3-group libraries determined and distributed by Newan and O'Brien. These and all other p-groups of order at most 2000 were constructed using the p-group generation algorithm. They were used to obtain all nilpotent groups of order at most 2000 in the catalogue.

The non-nilpotent groups of order at most 2000 in the library have been constructed by Besche and Eick using the methods of the GrpConst package of GAP 4. They rely on the catalogue of p-groups, the catalogue of perfect groups by Holt and Plesken, and the catalogue of irreducible soluble groups by Short and Höfling.

The groups of cubefree order at most 50000 have been computed by Besche and Eick using the GrpConst package and the work by Dietrich and Eick. They rely on the catalogue of irreducible soluble groups by Short and Höfling.

The groups of order p^4 have been constructed by Newman, the groups of order p^5 have been determined by Girnat and the groups of order p^6 have been classified by Newman, O'Brien and Vaughan-Lee.

The groups of order p^7 have been classified by O'Brien and Vaughan-Lee. The Small Groups Library includes them for p = 3,5,7,11.

The groups of order q^n * p for q and n fixed and p <> q have been constructed by Besche and Eick.

The groups of squarefree order and those of an order which factorises in at most 3 primes have been known for a long time.

Publications related to the development of the Small Groups Library

The following is intended to act as a reference point for further study of the algorithms employed and does not purport to be comprehensive; our algorithms rely on many other contributions to the field.

Hans Ulrich Besche and Bettina Eick
Construction of finite groups
J. Symbolic Comput. 27, 387 - 404 (1999)
Hans Ulrich Besche and Bettina Eick
The groups of order at most 1000 except 512 and 768
J. Symbolic Comput. 27, 405 - 413 (1999)
Hans Ulrich Besche and Bettina Eick
The groups of order q^n * p
Comm. Alg. 29, 1759 - 1772 (2001)
Hans Ulrich Besche, Bettina Eick and E.A. O'Brien
The groups of order at most 2000
Electron. Research Announc. Amer. Math. Soc. 7, 1 - 4 (2001)
Hans Ulrich Besche and Bettina Eick and E.A. O'Brien
A millennium project: constructing Small Groups
Internat. J. Algebra Comput. 12, 623 - 644 (2002)
Heiko Dietrich and Bettina Eick
Groups of cube-free order
J. Algebra. (To Appear)
Bettina Eick and E. A. O'Brien
The groups of order 512
Proceedings of the Abschlusstagung zum DFG Schwerpunkt Algorithmische Algebra und Zahlentheorie, Springer (1998), 379 - 380.
Bettina Eick and E. A. O'Brien
Enumerating p-groups
J. Austral. Math. Soc. 67, 191 - 205 (1999)
B. Girnat
Klassifikation der Gruppen bis zur Ordnung p^5
Staatsexamensarbeit, TU Braunschweig.
Otto Hölder
Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4
Math. Ann. 43, 301 - 412 (1893).
Rodney James, M.F. Newman and E.A. O'Brien
The groups of order 128
J. Algebra 129, 136-158 (1990)
M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee
Groups and nilpotent Lie rings whose order is the sixth power of a prime
J. Algebra 278, 383 - 401 (2003)
E.A. O'Brien
The p-group generation algorithm
J. Symbolic Comput. 9, 677 - 698 (1990)
E.A. O'Brien
The groups of order 256
J. Algebra 143, 219 - 235 (1991)
E. A. O'Brien and M. R. Vaughan-Lee
The groups of order p^7 for odd prime p.
J. Algebra 292, 243 - 258 (2005)

How to obtain the Small Groups library

The Small Groups library is available as part of the distribution of the computer algebra systems Gap and Magma. Both systems provide search facilities for the library and, for most available orders, an identification function corresponding to the groups in the library.

The latest version of the Small Groups library package for Gap is at current only available here: small.tar (16 MB).