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.
The notation
G ×
H stands for the
direct product of the two groups.
Order |
Groups |
1 |
C_{1} (the trivial group, abelian) |
2 |
C_{2} (abelian, simple) |
3 |
C_{3} (abelian, simple) |
4 |
C_{4} (abelian); C_{2} ×
C_{2} (abelian, isomorphic to the Klein four-group). |
5 |
C_{5} (abelian, simple) |
6 |
C_{6} (abelian); S_{3} (isomorphic
to D_{6}, the smallest non-abelian group) |
7 |
C_{7} (abelian, simple) |
8 |
C_{8} (abelian); C_{2} ×
C_{4} (abelian); C_{2} ×
C_{2} × C_{2} (abelian); D_{8};
Q_{8} (the quaternion group) |
9 |
C_{9} (abelian); C_{3} ×
C_{3} (abelian) |
10 |
C_{10} (abelian); D_{10} |
11 |
C_{11} (abelian, simple) |
12 |
C_{12} (abelian); C_{2} ×
C_{6} (abelian); D_{12}; A_{4};
the semidirect product of C_{3} and
C_{4}, where C_{4} acts on
C_{3} by inversion. |
13 |
C_{13} (abelian, simple) |
14 |
C_{14} (abelian); D_{14} |
15 |
C_{15} (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 .