The determination of all groups of a given order up to isomorphism is a central problem in finite group theory. It has been initiated in 1854 by A. Cayley who constructed the groups of order 4 and 6.

A large number of publications followed Cayley's work. For example,
Hall and Senior determined the groups of order `2 ^{n}` for

Then Newman and O'Brien introduced an algorithm to determine
groups of prime-power order, see OBr90. An implementation
of this method is available in the ANUPQ share package of GAP.
This method has been used to compute the groups of order `2 ^{n}` for

In this share package we introduce practical methods to determine up to isomorphism all groups of a given order. The algorithms are described in BE99. These methods have been used to construct the non-nilpotent groups of order at most 1000, see BE1000. The resulting catalogue of groups is available within the small groups library of GAP 4.

Our methods are not limited to groups of order at most 1000 and thus may be used to determine all or certain groups of higher order as well. However, it is not easy to say for which orders our methods are still practical and for which not. As a rule of thumb one can say that the number of primes and the size of the prime-powers contained in the factorisation of the given order determine the practicability of the algorithm; that is, the more primes are contained in the factorisation the more difficult the determination gets.

As an example, the construction of all non-nilpotent groups of order
`192 = 2 ^{6} cdot3` takes 17 minutes on an PC 400 Mhz. This is a medium
sized application of our methods. However, the construction of the groups
of order

Finally, we mention that the correctness of our algorithms is very hard to check for a user; in particular, since there are no other algorithms for the same purpose available, it might be difficult to verify that our methods compute all desired groups. Thus we note here that methods implemented in this share package have been used to compute large parts of the Small Groups library and this, in turn, has been checked by the authors as described in BE99 and BE1000.

Comments and suggestions on this share package are very welcome. Please send them to

centerline beick@tu-bs.de or hubesche@tu-bs.de.

Bug reports should also be e-mailed to either of these addresses.

grpconst manual

January 2020