GAP

The SmallGroups Library

This library has the status of an accepted GAP package, communicated in January 2002 by Mike F. Newman, Canberra.

Authors

Hans Ulrich Besche, Bettina Eick, and Eamonn O'Brien.

Description

The SmallGroups Library contains all groups of certain 'small' orders. The word 'Small' is used to mean orders less than a certain bound and orders whose prime factorisation is small in some sense. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. Currently, the library contains the following groups:

• those of order at most 2000 except 1024 (423 164 062 groups);
• those of cubefree order at most 50 000 (395 703 groups);
• those of order p7 for the primes p = 3,5,7,11 (907 489 groups);
• those of order pn for n <= 6 and all primes p;
• those of order qn * p where qn divides 28, 36, 55 or 74 and p is an arbitrary prime not equal to q;
• those of squarefree order;
• those whose order factorises into at most 3 primes.

The library also has an identification function: it returns the library number of a given group. Currently, this function is available for all orders in the library except for the orders 512 and 1536 and except for the orders p5, p6 and p7 above 2000.

The library is organised in 11 layers. Each layer contains the groups of certain orders and their corresponding group identification routines. It is possible to install the first n layers of the group library and the first m layers of the group identification for each 1 ≤ m ≤ n ≤ 11. In summary, the layers are:

• the groups whose order factorises into at most 3 primes.
• the remaining groups of order at most 1000 except 512 and 768.
• the remaining groups of order 2n * p with n < 9 and p an odd prime.
• the groups of order 55 and 74 and the remaining groups of order qn * p where qn divides 36, 55 or 74 and p is a prime with p not equal to q.
• the remaining groups of order at most 2000 except 1024, 1152, 1536 and 1920.
• the groups of orders 1152 and 1920.
• the groups of order 512.
• the groups of order 1536.
• the remaining groups of order p4, p5 and p6.
• the remaining groups of squarefree order and of cubefree order at most 50000.
• the remaining groups of order p7 for p = 3,5,7,11.

Hans Ulrich Besche
Institut Computational Mathematics
Universität Braunschweig
Pockelsstr. 14
38106 Braunschweig
Germany
email: hubesche@tu-bs.de

Bettina Eick
Institut Computational Mathematics
TU Braunschweig
Pockelsstr. 14
D-38106 Braunschweig
Germany
email: beick@tu-bs.de

Eamonn O'Brien
Department of Mathematics
University of Auckland
Auckland, Private Bag 92019
New Zealand
email: obrien@math.auckland.ac.nz