List of small groups

This is a list of small finite mathematical 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.

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)

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 .