List of small groups

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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.

1 Glossary
2 List of small non-abelian groups
3 Combined list
4 Small groups library
5 See also
6 External links

[edit] Glossary

Z_n: the cyclic group of order n (often the notation C_n is used, or Z / n Z).
Dih_n: the dihedral group of order 2n (often the notation D_n is used, and sometimes D_2n )
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 notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

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 Z_n, where n is prime.) We use the equality sign ("=") to denote isomorphism.

The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of subgroups the trivial group and the group itself are not listed.

[edit] List of small non-abelian groups

See also the list of small abelian groups and the combined list below.

Note that e.g. "3 × Z₂" means that there are 3 subgroups of type Z₂ (NOT a left coset of Z₂), while elsewhere the cross means direct product.

Order	Group	Subgroups	Properties
6	S₃ = Dih₃	Z₃ , 3 × Z₂	the smallest non-abelian group
8	Dih₄	Z₄, 2 × Dih₂ , 5 × Z₂	non-abelian
8	Quaternion group, Q₈ = Dic₂	3 × Z₄ , Z₂	non-abelian; the smallest Hamiltonian group
10	Dih₅	Z₅ , 5 × Z₂	non-abelian
12	Dih₆ = Dih₃ × Z₂	Z₆ , 2 × Dih₃ , 3 × Dih₂ , Z₃ , 7 × Z₂	non-abelian	edit
	A₄	Z₂², 4 × Z₃, 3 × Z₂	non-abelian; smallest group demonstrating that the converse of Lagrange's theorem is not true: no subgroup of order 6
	Dic₃ = the semidirect product of Z₃ and Z₄, where Z₄ acts on Z₃ by inversion	Z₂, Z₃, 3 × Z₄, Z₆	non-abelian
14	Dih₇	Z₇ , 7 × Z₂	non-abelian
16	Dih₈	Z₈ , 2 × Dih₄ , 4 × Dih₂ , Z₄ , 9 × Z₂	non-abelian	edit
	Dih₄ × Z₂	2 × Dih₄ , Z₄ × Z₂ , 2 × Z₂^{3^{, 7 × Z₂² , 2 × Z₄ , 11 × Z₂}}	non-abelian
	Generalized quaternion group, Q₁₆ = Dic₄		non-abelian
	Q₈ × Z₂		non-abelian, Hamiltonian
	The order 16 quasidihedral group		non-abelian
	The order 16 modular group		non-abelian
	The semidirect product of Z₄ and Z₄ 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

[edit] Combined list

Order	Group	Subgroups	Properties
1	trivial group = Z₁ = S₁ = A₂	-	abelian; this and various other properties hold trivially
2	Z₂ = S₂ = Dih₁	-	abelian, simple, the smallest non-trivial group
3	Z₃ = A₃	-	abelian, simple
4	Z₄	Z₂	abelian
4	Klein four-group = Z₂ × Z₂ = Dih₂	3 × Z₂	abelian, the smallest non-cyclic group
5	Z₅	-	abelian, simple
6	Z₆ = Z₂ × Z₃	Z₂ , Z₃	abelian
6	S₃ = Dih₃	Z₃ , 3 × Z₂	the smallest non-abelian group
7	Z₇	-	abelian, simple
8	Z₈	Z₄ , Z₂	abelian
	Z₂ ×Z₄	2 × Z₄ , 3 ×Z₂ , Dih₂	abelian
	Z₂ × Z₂ × Z₂ = Dih₂ × Z₂	7 × Z₂ × Z₂ , 7 × Z₂	abelian
	Dih₄	Z₄, 2 × Dih₂ , 5 × Z₂	non-abelian
	Quaternion group, Q₈ = Dic₂	3 × Z₄ , Z₂	non-abelian; the smallest Hamiltonian group
9	Z₉	Z₃	abelian
9	Z₃ × Z₃	4 × Z₃	abelian
10	Z₁₀ = Z₂ × Z₅	Z₅ , Z₂	abelian
10	Dih₅	Z₅ , 5 × Z₂	non-abelian
11	Z₁₁	-	abelian, simple
12	Z₁₂ = Z₄ × Z₃	Z₆ , Z₄ , Z₃ , Z₂	abelian
	Z₂ × Z₆ = Z₂ × Z₂ × Z₃ = Dih₂ × Z₃	3 × Z₆, Z₃, Dih₂, 3 × Z₂	abelian
	Dih₆ = Dih₃ × Z₂	Z₆ , 2 × Dih₃ , 3 × Dih₂ , Z₃ , 7 × Z₂	non-abelian	edit
	A₄	Z₂², 4 × Z₃, 3 × Z₂	non-abelian; smallest group demonstrating that the converse of Lagrange's theorem is not true: no subgroup of order 6
	Dic₃ = the semidirect product of Z₃ and Z₄, where Z₄ acts on Z₃ by inversion	Z₂, Z₃, 3 × Z₄, Z₆	non-abelian
13	Z₁₃	-	abelian, simple
14	Z₁₄ = Z₂ × Z₇	Z₇ , Z₂	abelian
14	Dih₇	Z₇ , 7 × Z₂	non-abelian
15	Z₁₅ = Z₃ × Z₅	Z₅ , Z₃	abelian
16	Z₁₆	Z₈ , Z₄ , Z₂	abelian
	Z₂⁴	15 × Z₂, 35 × Dih₂, 15 × Z₂³	abelian
	Z₄ × Z₂²	7 × Z₂, 4 × Z₄, 7 × Dih₂, Z₂³, 6 × Z₄ × Z₂	abelian
	Z₈ × Z₂	3 × Z₂, 2 × Z₄, Dih₂, 2 × Z₈, Z₄ × Z₂	abelian
	Z₄²	3 × Z₂, 6 × Z₄, Dih₂, 3 × Z₄ × Z₂	abelian
	Dih₈	Z₈ , 2 × Dih₄ , 4 × Dih₂ , Z₄ , 9 × Z₂	non-abelian	edit
	Dih₄ × Z₂	2 × Dih₄ , Z₄ × Z₂ , 2 × Z₂^{3^{, 7 × Z₂² , 2 × Z₄ , 11 × Z₂}}	non-abelian
	Generalized quaternion group, Q₁₆ = Dic₄		non-abelian
	Q₈ × Z₂		non-abelian, Hamiltonian
	The order 16 quasidihedral group		non-abelian
	The order 16 modular group		non-abelian
	The semidirect product of Z₄ and Z₄ 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

[edit] 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.