BIGpedia - List of small groups - Encyclopedia and Dictionary Online

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.
Dic_n: the dicyclic group of order 4n.

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.

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

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,
4	Klein four-group = C₂ × C₂ = D₄	abelian, the smallest non-cyclic group
5	C₅	abelian, simple
6	C₆ = C₂ × C₃	abelian
6	S₃ = D₆	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 = Dic₂	non-abelian; the smallest Hamiltonian group
9	C₉	abelian
9	C₃ × C₃	abelian
10	C₁₀ = C₂ × C₅	abelian
10	D₁₀	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
	Dic₃ = 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
14	D₁₄	non-abelian
15	C₁₅ = C₃ × C₅	abelian
16	C₁₆	abelian
	C₂ × C₂ × C₂ × C₂	abelian
	C₂ × C₂ × C₄	abelian
	C₂ × C₈	abelian
	C₄ × C₄	abelian
	D₁₆	non-abelian
	Generalized quaternion group, Q₁₆ = Dic₄	non-abelian
	C₂ × D₈	non-abelian
	C₂ × Q₈	non-abelian
	The order 16 quasidihedral group	non-abelian
	The order 16 modular group	non-abelian
	The semidirect product of C₄ and C₄ where one factor acts on the other by inversion	non-abelian
	The group generated by the Pauli matrices	non-abelian
	G_4,4	non-abelian

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 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 .

Categories: Group theory | Computational group theory