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.

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 are 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.

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	Cycle graph
6	S3 = Dih3	Z3 , 3 × Z2	the smallest non-abelian group
8	Dih4	Z4, 2 × Dih2 , 5 × Z2	non-abelian
	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 × Z23, 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
	G4,4		non-abelian

Combined list

Order	Group	Subgroups	Properties	Cycle graph
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
	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
	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
	Z3 × Z3	4 × Z3	abelian
10	Z10 = Z2 × Z5	Z5 , Z2	abelian
	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
	Dih7	Z7 , 7 × Z2	non-abelian
15	Z15 = Z3 × Z5	Z5 , Z3	abelian
16	Z16	Z8 , Z4 , Z2	abelian
	Z24	3 × Z2, 6 × Z4, Dih2, 3 × Z4 × Z2	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	15 × Z2, 35 × Dih2, 15 × Z23	abelian
	Dih8	Z8 , 2 × Dih4 , 4 × Dih2 , Z4 , 9 × Z2	non-abelian		[edit]
	Dih4 × Z2	2 × Dih4 , Z4 × Z2 , 2 × Z23, 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
	G4,4		non-abelian

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 .

See also

• Small Latin squares and quasigroups

External links

• [Small groups]

 

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