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. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 24
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 24
subgroup structure	subgroup structure of groups of order 24
linear representation theory	linear representation theory of groups of order 24
endomorphism structure, automorphism structure	endomorphism structure of groups of order 24

Statistics at a glance

The number 24 has prime factorization $24=2^3\cdot 3$. There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^a{q}^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity	Value	Explanation
Total number of groups	15
Number of abelian groups	3	(number of abelian groups of order $2^3$) times (number of abelian groups of order $3^1$) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 1) = $3\times 1=3$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups	5	(number of groups of order 8) times (number of groups of order 3) = $5\times 1=5$. See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which follows from equivalence of definitions of finite nilpotent group.
Number of solvable groups	15	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^a{q}^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups	0	Follows from all groups of this order being solvable

The list

There are 15 groups of order 24.

Group	Second part of GAP ID (ID is (24,second part))	Nilpotency class	Derived length
nontrivial semidirect product of Z3 and Z8	1	not nilpotent	2
cyclic group:Z24	2	1	1
special linear group:SL(2,3)	3	not nilpotent	3
dicyclic group:Dic24	4	not nilpotent	2
direct product of S3 and Z4	5	not nilpotent	2
dihedral group:D24	6	not nilpotent	2
direct product of Dic12 and Z2	7	not nilpotent	2
SmallGroup(24,8)	8	not nilpotent	2
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	1
direct product of D8 and Z3	10	2	2
direct product of Q8 and Z3	11	2	2
symmetric group:S4	12	not nilpotent	3
direct product of A4 and Z2	13	not nilpotent	2
direct product of D12 and Z2 (also direct product of S3 and V4)	14	not nilpotent	2
direct product of E8 and Z3	15	1	1

Sylow subgroups

2-Sylow subgroups

Here is the occurrence summary:

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

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

GAP implementation

The order 24 is part of GAP's SmallGroup library. Hence, any group of order 24 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 24 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(24);

  There are 15 groups of order 24.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 6, 1 ].
     2 has Frattini factor [ 6, 2 ].
     3 has Frattini factor [ 12, 3 ].
     4 - 8 have Frattini factor [ 12, 4 ].
     9 - 11 have Frattini factor [ 12, 5 ].
     12 - 15 have trivial Frattini subgroup.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
     LGLength, FrattinifactorSize and FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.