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.
The library also contains the number of groups of order 1024 and of order $p^7$ for arbitrary primes $p$. The groups of order $3^8$ and those of order $p^7$ for $p$ an arbitrary prime are also available if the SglPPow package and the LiePRing Package of GAP are loaded.

The Small Groups library has been prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien. The groups in the Small Groups library have been been constructed by the authors of the library with the help of Mike Newman (orders 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
A table of numbers of groups of small order
References to the construction of the Small Groups library
How to obtain and use the Small Groups library
A 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 Newman 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 based on the work by Dietrich and Eick. They rely on the catalogue of irreducible soluble groups by Short and Höfling.

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, of an order which factorises in at most 3 primes and of order dividing $p^4$ 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 292, 122 - 137 (2005)

  • 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.