# List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below. If you know whether G is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.

## Glossary

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.) We use the equality sign ("=") to denote isomorphism.

The identity element in the cycle graphs are represented by the black circle.

## List

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

## Small groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which 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 for order 1024 (423 164 062 groups);
• those of order 55 and 74 (92 groups);
• those of order qn×p where qn divides 28, 36, 55 or 74 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 .

Order Group Properties Cycle graph
1 trivial group = C1 = S1 = A2 abelian
Missing image
GroupDiagramMiniC1.png

2 C2 = S2 abelian, simple, the smallest non-trivial group
Missing image
GroupDiagramMiniC2.png

3 C3 = A3 abelian, simple
Missing image
GroupDiagramMiniC3.png

4 C4 abelian,
Missing image
GroupDiagramMiniC4.png

Klein four-group = C2 × C2 = D4 abelian, the smallest non-cyclic group
Missing image
GroupDiagramMiniD4.png

5 C5 abelian, simple
Missing image
GroupDiagramMiniC5.png

6 C6 = C2 × C3 abelian
Missing image
GroupDiagramMiniC6.png

S3 = D6 the smallest non-abelian group
Missing image
GroupDiagramMiniD6.png

7 C7 abelian, simple
Missing image
GroupDiagramMiniC7.png

8 C8 abelian
Missing image
GroupDiagramMiniC8.png

C2 ×C4 abelian
Missing image
GroupDiagramMiniC2C4.png

C2 × C2 × C2 = D4 × C2 abelian
Missing image
GroupDiagramMiniC2x3.png

D8 non-abelian
Quaternion group, Q8 = Dic2 non-abelian; the smallest Hamiltonian group
9 C9 abelian
Missing image
GroupDiagramMiniC9.png

C3 × C3 abelian
Missing image
GroupDiagramMiniC3x2.png

10 C10 = C2 × C5 abelian
Missing image
GroupDiagramMiniC10.png

D10 non-abelian
Missing image
GroupDiagramMiniD10.png

11 C11 abelian, simple
Missing image
GroupDiagramMiniC11.png

12 C12 = C4 × C3 abelian
Missing image
GroupDiagramMiniC12.png

C2 × C6 = C2 × C2 × C3 = D4 × C3 abelian
Missing image
GroupDiagramMiniC2C6.png

D12 = D6 × C2 non-abelian
Missing image
GroupDiagramMiniD12.png

A4 non-abelian
Missing image
GroupDiagramMiniA4.png

Dic3 = the semidirect product of C3 and C4, where C4 acts on C3 by inversion non-abelian
Missing image
GroupDiagramMiniX12.png

13 C13 abelian, simple
14 C14 = C2 × C7 abelian
Missing image
GroupDiagramMiniC14.png

D14 non-abelian
Missing image
GroupDiagramMiniD14.png

15 C15 = C3 × C5 abelian
Missing image
GroupDiagramMiniC15.png

16 C16 abelian
Missing image
GroupDiagramMiniC16.png

C2 × C2 × C2 × C2 abelian
C2 × C2 × C4</sup> abelian
Missing image
GroupDiagramMiniC2x2C4.png

C2 × C8 abelian
Missing image
GroupDiagramMiniC2C8.png

C4 × C4 abelian
Missing image
GroupDiagramMiniC4x2.png

D16 non-abelian
Missing image
GroupDiagramMiniD16.png

Generalized quaternion group, Q16 = Dic4 non-abelian
Missing image
GroupDiagramMiniQ16.png

C2 × D8 non-abelian
Missing image
GroupDiagramMiniC2D8.png

C2 × Q8 non-abelian
Missing image
GroupDiagramMiniC2Q8.png

The order 16 quasidihedral group non-abelian
Missing image
GroupDiagramMiniQH16.png

The order 16 modular group non-abelian
Missing image
GroupDiagramMiniMOD16.png

The semidirect product of C4 and C4 where one factor acts on the other by inversion non-abelian
Missing image
GroupDiagramMinix3.png

The group generated by the Pauli matrices non-abelian
Missing image
GroupDiagramMiniPauli.png

G4,4 non-abelian
Missing image
GroupDiagramMiniG44.png

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