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
The notation G × H stands for the direct product of the two groups.
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
C4, where C4 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.
It contains explicit descriptions of the available groups in computer readable format.
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 .