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 projective representation theory of groups of order 24 modular representation theory of groups of order 24
endomorphism structure, automorphism structure	endomorphism structure of groups of order 24
group cohomology	group cohomology of groups of order 24

Statistics at a glance

To understand these in a broader context, see groups of order 3.2^n

Factorization and useful forms

The number 24 has prime factors 2 and 3 and prime factorization:

$24=2^{3}\cdot 3^{1}=8\cdot 3$

Other expressions for this number are:

$24=4!=3^{3}-3=2^{2}(2^{2}-1)(2^{2}-2)=2(2^{2})(2^{2}-1)=2^{3}\cdot 3$

Group counts

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 up to isomorphism	15
Number of abelian groups (i.e., finite abelian groups) up to isomorphism	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 (i.e., finite nilpotent groups) up to isomorphism	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 in turn follows from equivalence of definitions of finite nilpotent group.
Number of solvable groups (i.e., finite solvable groups) up to isomorphism	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 possible multisets of composition factors, i.e., number of equivalence classes under composition factor-equivalence	1	Follows from all groups of this order being solvable. The only possibility is: cyclic group:Z2 (3 times), cyclic group:Z3 (1 time). See order of group is product of orders of composition factors and classification of possible multisets of composition factors for groups of a given 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	2-Sylow subgroup	Is the 2-Sylow subgroup normal?	Is the 3-Sylow subgroup normal?
nontrivial semidirect product of Z3 and Z8	1	not nilpotent	2	cyclic group:Z8	No	Yes
cyclic group:Z24	2	1	1	cyclic group:Z8	Yes	Yes
special linear group:SL(2,3)	3	not nilpotent	3	quaternion group	Yes	No
dicyclic group:Dic24	4	not nilpotent	2	quaternion group	No	Yes
direct product of S3 and Z4	5	not nilpotent	2	direct product of Z4 and Z2	No	Yes
dihedral group:D24	6	not nilpotent	2	dihedral group:D8	No	Yes
direct product of Dic12 and Z2	7	not nilpotent	2	direct product of Z4 and Z2	No	Yes
SmallGroup(24,8)	8	not nilpotent	2	dihedral group:D8	No	Yes
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	1	direct product of Z4 and Z2	Yes	Yes
direct product of D8 and Z3	10	2	2	dihedral group:D8	Yes	Yes
direct product of Q8 and Z3	11	2	2	quaternion group	Yes	Yes
symmetric group:S4	12	not nilpotent	3	dihedral group:D8	No	No
direct product of A4 and Z2	13	not nilpotent	2	elementary abelian group:E8	Yes	No
direct product of D12 and Z2 (also direct product of S3 and V4)	14	not nilpotent	2	elementary abelian group:E8	No	Yes
direct product of E8 and Z3	15	1	1	elementary abelian group:E8	Yes	Yes

Relation with other orders

Divisors of the order

More in-depth information can be found under subgroup structure of groups of order 24.

Divisor	Quotient value	Number of groups of the order	Information on groups of the order
2	12	1	cyclic group:Z2
3	8	1	cyclic group:Z3
4	6	2	groups of order 4
6	4	2	groups of order 6
8	3	5	groups of order 8
12	2	5	groups of order 12

Multiples of the order

More in-depth information can be found under supergroups of groups of order 24.

Multiplier (other factor)	Multiple	Number of groups	Information on groups of the order
2	48	52	groups of order 48
3	72	50	groups of order 72
4	96	231	groups of order 96
5	120	47	groups of order 120
6	144	197	groups of order 144
7	168	57	groups of order 168
8	192	1543	groups of order 192
9	216	177	groups of order 216
10	240	208	groups of order 240

Similar prime structure

See classification of groups of order eight times a prime congruent to 3 mod 8, or more generally classification of groups of order eight times a prime.

Minimal order attaining number

$24$ is the smallest number such that there are precisely $15$ groups of that order up to isomorphism. That is, the value of the minimal order attaining function at $15$ is $24$ .

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.