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
- C_{n}: the cyclic group of order n
- D_{n}: the dihedral group of order n
- S_{n}: the symmetric group of degree n, containing the n permutations of n elements.
- A_{n}: the alternating group of degree n, containing the n!/2 even permutations of n elements.
List
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) |
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 .