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List of small groups
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List of small groups

In mathematics, the finite groups of small order can be listed up to group isomorphism.


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.)


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 abelian
D8 non-abelian
Quaternion group = Q8 the smallest non-abelian group, each of whose subgroups is normal
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 abelian
D12 non-abelian
A4 non-abelian
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 (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: 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 .