List of groups of small order

The following is a list of the finite groups of smallest order.

Group	Order	GAP ID (second part)	Abelian?
Trivial group	1	1	Yes
Cyclic group:Z2	2	1	Yes
Cyclic group:Z3	3	1	Yes
Cyclic group:Z4	4	1	Yes
Klein four-group	4	2	Yes
Cyclic group:Z5	5	1	Yes
Symmetric group:S3	6	2	No
Cyclic group:Z6	6	2	Yes
Cyclic group:Z7	7	1	Yes
Cyclic group:Z8	8	1	Yes
Direct product of Z4 and Z2	8	2	Yes
Dihedral group:D8	8	3	No
Quaternion group	8	4	No
Elementary abelian group:E8	8	5	Yes
Cyclic group:Z9	9	1	Yes
Elementary abelian group:E9	9	2	Yes
Dihedral group:D10	10	1	No
Cyclic group:Z10	10	2	Yes
Cyclic group:Z11	11	1	Yes
Dicyclic group:Dic12	12	1	No
Cyclic group:Z12	12	2	Yes
Alternating group:A4	12	3	No
Dihedral group:D12	12	4	No
Direct product of Z6 and Z2	12	5	Yes
Cyclic group:Z13	13	1	Yes
Dihedral group:D14	14	1	No
Cyclic group:Z14	14	2	Yes
Cyclic group:Z15	15	1	Yes
cyclic group:Z16	16	1	Yes
direct product of Z4 and Z4	16	2	Yes
SmallGroup(16,3)	16	3	No
nontrivial semidirect product of Z4 and Z4	16	4	No
direct product of Z8 and Z2	16	5	Yes
modular maximal-cyclic group:M16	16	6	No
dihedral group:D16	16	7	No
semidihedral group:SD16	16	8	No
generalized quaternion group:Q16	16	9	No
direct product of Z4 and V4	16	10	Yes
direct product of D8 and Z2	16	11	No
direct product of Q8 and Z2	16	12	No
central product of D8 and Z4	16	13	No
elementary abelian group:E16	16	14	Yes
Cyclic group:Z17	17	1	Yes
dihedral group:D18	18	1	No
cyclic group:Z18	18	2	Yes
direct product of S3 and Z3	18	3	No
generalized dihedral group for E9	18	4	No
direct product of Z6 and Z3	18	5	Yes
Cyclic group:Z19	19	1	Yes
dicyclic group:Dic20	20	1	No
cyclic group:Z20	20	2	Yes
general affine group:GA(1,5)	20	3	No
dihedral group:D20	20	4	No
direct product of Z10 and Z2	20	5	Yes
General semilinear group:GammaL(1,8)	21	1	No
cyclic group:Z21	21	2	Yes
dihedral group:D22	22	1	No
cyclic group:Z22	22	2	Yes
Cyclic group:Z23	23	1	Yes
nontrivial semidirect product of Z3 and Z8	24	1	No
Cyclic group:Z24	24	2	Yes
special linear group:SL(2,3)	24	3	No
dicyclic group:Dic24	24	4	No
direct product of S3 and Z4	24	5	No
dihedral group:D24	24	6	No
direct product of Dic12 and Z2	24	7	No
SmallGroup(24,8)	24	8	No
direct product of Z6 and Z4	24	9	Yes
direct product of D8 and Z3	24	10	No
direct product of Q8 and Z3	24	11	No
symmetric group:S4	24	12	No
direct product of A4 and Z2	24	13	No
direct product of D12 and Z2	24	14	No
direct product of E8 and Z3	24	15	Yes