The List of small groups reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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.

The notation G × H stands for the direct product of the two groups. Non-abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups C_n, where n is prime.)

List

Order	Group
1	C1 (the trivial group)
2	C2 (the smallest non-trivial simple group)
3	C3 (simple)
4	C4; C2 × C2 (isomorphic to the Klein four-group, the smallest non-cyclic group)
5	C5 (simple)
6	C6; S3 (isomorphic to D6, the smallest non-abelian group)
7	C7 (simple)
8	C8; C2 × C4; C2 × C2 × C2; D8 (the octic group, non-abelian); Q8 (the quaternion group, non-abelian)
9	C9; C3 × C3
10	C10; D10 (non-abelian)
11	C11 (simple)
12	C12; C2 × C6; D12 (non-abelian); A4 (non-abelian); the semidirect product of C3 and C4, where C4 acts on C3 by inversion (non-abelian)
13	C13 (simple)
14	C14; D14 (non-abelian)
15	C15

Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)

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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 .

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