Groups of order 24

This article gives information about, and links to more details on, groups of order 24
See pages on algebraic structures of order 24 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 24.

Statistics at a glance

Quantity	Value
Total number of groups	15
Number of abelian groups	3
Number of nilpotent groups	5
Number of solvable groups	15
Number of simple groups	0

The list

There are 15 groups of order 24.

Group	Second part of GAP ID (ID is (24,second part))
nontrivial semidirect product of Z3 and Z8	1
cyclic group:Z24	2
special linear group:SL(2,3)	3
dicyclic group:Dic24	4
direct product of S3 and Z4	5
dihedral group:D24	6
direct product of Dic12 and Z2	7
SmallGroup(24,8)	8
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9
direct product of D8 and Z3	10
direct product of Q8 and Z3	11
symmetric group:S4	12
direct product of A4 and Z2	13
direct product of D12 and Z2 (also direct product of S3 and V4)	14
direct product of E8 and Z3	15

Sylow subgroups

2-Sylow subgroups

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

Group	Second part of GAP ID (ID is (24,second part))	2-Sylow subgroup	Second part of GAP ID	Number of 2-Sylow subgroups
nontrivial semidirect product of Z3 and Z8	1	cyclic group:Z8	1	3
cyclic group:Z24	2	cyclic group:Z8	1	1
special linear group:SL(2,3)	3	quaternion group	4	1
dicyclic group:Dic24	4	quaternion group	4	3
direct product of S3 and Z4	5	direct product of Z4 and Z2	2	3
dihedral group:D24	6	dihedral group:D8	3	3
direct product of Dic12 and Z2	7	direct product of Z4 and Z2	2	3
SmallGroup(24,8)	8	dihedral group:D8	3	3
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	direct product of Z4 and Z2	2	1
direct product of D8 and Z3	10	dihedral group:D8	3	1
direct product of Q8 and Z3	11	quaternion group	4	1
symmetric group:S4	12	dihedral group:D8	3	3
direct product of A4 and Z2	13	elementary abelian group:E8	5	1
direct product of D12 and Z2 (also direct product of S3 and V4)	14	elementary abelian group:E8	5	3
direct product of E8 and Z3	15	elementary abelian group:E8	5	1

3-Sylow subgroups

Note that the 3-Sylow subgroup is isomorphic to cyclic group:Z3 in all cases. By the congruence condition on Sylow numbers as well as the divisibility condition on Sylow numbers, the only possibilities for the number of 3-Sylow subgroups is 1 or 4. In the former case, we have a normal Sylow subgroup. In the latter case, the normalizer of the Sylow subgroup has order 6, and is thus either cyclic group:Z6 or symmetric group:S3.

Group	Second part of GAP ID (ID is (24,second part))	Number of 3-Sylow subgroups	Normalizer of Sylow subgroup
nontrivial semidirect product of Z3 and Z8	1	1	whole group
cyclic group:Z24	2	1	whole group
special linear group:SL(2,3)	3	4	cyclic group:Z6
dicyclic group:Dic24	4	1	whole group
direct product of S3 and Z4	5	1	whole group
dihedral group:D24	6	1	whole group
direct product of Dic12 and Z2	7	1	whole group
SmallGroup(24,8)	8	1	whole group
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	whole group
direct product of D8 and Z3	10	1	whole group
direct product of Q8 and Z3	11	1	whole group
symmetric group:S4	12	4	symmetric group:S3
direct product of A4 and Z2	13	4	cyclic group:Z6
direct product of D12 and Z2 (also direct product of S3 and V4)	14	1	whole group
direct product of E8 and Z3	15	1	whole group