The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below. If you know whether G is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.
The notation G × H stands for the direct product of the two groups. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Cn, where n is prime.) We use the equality sign ("=") to denote isomorphism.
Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)
The group theoretical computer algebra systemGAP contains the "Small Groups library" which 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 for order 1024 (423 164 062 groups);
those of order 5^5 and 7^4 (92 groups);
those of order q^n * p where q^n divides 2^8, 3^6, 5^5 or 7^4 and p is an arbitrary prime which differs from q;
those whose order factorises into at most 3 primes.
It contains explicit descriptions of the available groups in computer readable format.
This group is isomorphic to the group of integers (modulo
In general, a group action is when a group acts on a set, permuting its elements, so that the map from the group to the permutation group of the set is a homomorphism.
Group actions, and in particular representations, are very important in applications, not only to group theory, but also to physics and chemistry.
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