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

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

List

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 = D3 × C2	the smallest non-abelian group
7	C7	abelian, simple
8	C8	abelian
	C2 × C4	abelian
	C2 × C2 × C2 = D4 × C2	abelian
	D8	non-abelian
	Quaternion group = Q8	non-abelian; the smallest Hamiltonian group
9	C9	abelian
	C3 × C3	abelian
10	C10 = C2 × C5	abelian
	D10 = D5 × C2	non-abelian
11	C11	abelian, simple
12	C12 = C4 × C3	abelian
	C2 × C6 = C2 × C2 × C3 = D4 × C3	abelian
	D12 = D6 × C2	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 = D7 × C2	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.)

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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 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 .