List of small groups
In mathematical, this is a list of small finite groups. For each order, all groups of that order up to group isomorphism are listed.Glossary
- Cn: the cyclic group of order n
- Dn: the dihedral group of order n
- Sn: the symmetric group of degree n, containing the n permutations of n elements.
- An: the alternating group of degree n, containing the n!/2 even permutations of n elements.
List
| Order | Groups |
|---|---|
| 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 C'\'4, where C''4 acts on C3 by inversion. |
| 13 | C13 (abelian, simple) |
| 14 | C14 (abelian); D14 |
| 15 | C15 (abelian) |
- 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 .
