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The following list in mathematics contains the finite groups of small order up to group isomorphism.
The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below. If you know whether G is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.
Glossary
The notation G × H stands for the direct product of the two groups. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Cn, where n is prime.) We use the equality sign ("=") to denote isomorphism.
List
| Order |
Group |
Properties |
| 1 |
trivial group = C1 = S1 = A2 |
abelian |
| 2 |
C2 = S2 |
abelian, simple, the smallest non-trivial group |
| 3 |
C3 = A3 |
abelian, simple |
| 4 |
C4 |
abelian, |
| Klein four-group = C2 ×
C2 = D4 |
abelian, the smallest non-cyclic group |
| 5 |
C5 |
abelian, simple |
| 6 |
C6 = C2 × C3 |
abelian |
| S3 = D6 |
the smallest non-abelian group |
| 7 |
C7 |
abelian, simple |
| 8 |
C8 |
abelian |
| C2 ×
C4 |
abelian |
| C2 ×
C2 × C2 = D4 × C2 |
abelian |
|
D8 |
non-abelian |
| Quaternion group = Q8 |
non-abelian; the smallest Hamiltonian group |
| 9 |
C9 |
abelian |
| C3 ×
C3 |
abelian |
| 10 |
C10 = C2 × C5 |
abelian |
| D10 |
non-abelian |
| 11 |
C11 |
abelian, simple |
| 12 |
C12 = C4 × C3 |
abelian |
| C2 × C6 = C2 ×
C2 × C3 = D4 × C3 |
abelian |
| D12 = D6 × C2 |
non-abelian |
| A4 |
non-abelian |
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the semidirect product of C3 and
C4, where C4 acts on
C3 by inversion |
non-abelian |
| 13 |
C13 |
abelian, simple |
| 14 |
C14 = C2 × C7 |
abelian |
| D14 |
non-abelian |
| 15 |
C15 = C3 × C5 |
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 contains the "Small Groups library" which 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 for order 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 .
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