List of small groups
In mathematics, the finite groups of small order can be listed up to group isomorphism.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 | Group | Properties |
---|---|---|
1 | trivial group = C_{1} = S_{1} = A_{2} | abelian |
2 | C_{2} = S_{2} | abelian, simple, the smallest non-trivial group |
3 | C_{3} = A_{3} | abelian, simple |
4 | C_{4} | abelian, |
Klein four-group = C_{2} × C_{2} = D_{4} | abelian, the smallest non-cyclic group | |
5 | C_{5} | abelian, simple |
6 | C_{6} = C_{2} × C_{3} | abelian |
S_{3} = 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} | non-abelian | |
Quaternion group = Q_{8} | the smallest non-abelian group, each of whose subgroups is normal | |
9 | C_{9} | abelian |
C_{3} × C_{3} | abelian | |
10 | C_{10} = C_{2} × C_{5} | abelian |
D_{10} | non-abelian | |
11 | C_{11} | abelian, simple |
12 | C_{12} = C_{4} × C_{3} | abelian |
C_{2} × C_{6} = C_{2} × C_{2} × C_{3} | abelian | |
D_{12} | non-abelian | |
A_{4} | non-abelian | |
the semidirect product of C_{3} and C_{4}, where C_{4} acts on C_{3} by inversion | non-abelian | |
13 | C_{13} | abelian, simple |
14 | C_{14} = C_{2} × C_{7} | abelian |
D_{14} | non-abelian | |
15 | C_{15} = C_{3} × C_{5} | 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 .