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.

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)

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.)

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 .