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# List of small groups

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This is a list of small finite mathematical 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)

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 .

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