In mathematical, this is a list of small finite groups. For each order, all groups of that order up to group isomorphism are listed.
|1||C1 (the trivial group, abelian)|
|2||C2 (abelian, simple)|
|3||C3 (abelian, simple)|
|4||C4 (abelian); C2 × C2 (abelian, isomorphic to the Klein four-group).|
|5||C5 (abelian, simple)|
|6||C6 (abelian); S3 (isomorphic to D6, the smallest non-abelian group)|
|7||C7 (abelian, simple)|
|8||C8 (abelian); C2 × C4 (abelian); C2 × C2 × C2 (abelian); D8; Q8 (the quaternion group)|
|9||C9 (abelian); C3 × C3 (abelian)|
|10||C10 (abelian); D10|
|11||C11 (abelian, simple)|
|12||C12 (abelian); C2 × C6 (abelian); D12; A4; the semidirect product of C3 and C4, where C4 acts on C3 by inversion.|
|13||C13 (abelian, simple)|
|14||C14 (abelian); D14|
- Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)
The group theoretical computer algebra system GAP (available for free at http://www.gap-system.org/ ) contains the "Small Groups library": it 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 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.
The library has been constructed and prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien; see http://www.tu-bs.de/~hubesche/small.html .