# Difference between revisions of "Groups of order 24"

See pages on algebraic structures of order 24| See pages on groups of a particular order

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 $p^aq^b$-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 $p^aq^b$-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
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
semidirect product of Z3 and D8 with action kernel V4 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

## 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 Relationship (subgroup perspective) Relationship (quotient value)
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 Relationship (subgroup perspective) Relationship (quotient perspective)
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

## 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.