# Linear representation theory of groups of order 24

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 24.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 24

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

### Full listing

Group Second part of GAP ID (ID is (24,second part)) Nilpotency class Derived length Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Total number of irreps
nontrivial semidirect product of Z3 and Z8 1 not nilpotent 2 8 4 0 12
cyclic group:Z24 2 1 1 24 0 0 24
special linear group:SL(2,3) 3 not nilpotent 3 3 3 1 7
dicyclic group:Dic24 4 not nilpotent 2 4 5 0 9
direct product of S3 and Z4 5 not nilpotent 2 8 4 0 12
dihedral group:D24 6 not nilpotent 2 4 5 0 9
direct product of Dic12 and Z2 7 not nilpotent 2 8 4 0 12
SmallGroup(24,8) 8 not nilpotent 2 4 5 0 9
direct product of Z6 and Z4 (also, direct product of Z12 and Z2) 9 1 1 24 0 0 24
direct product of D8 and Z3 10 2 2 12 3 0 15
direct product of Q8 and Z3 11 2 2 12 3 0 15
symmetric group:S4 12 not nilpotent 3 2 1 2 5
direct product of A4 and Z2 13 not nilpotent 2 6 0 2 8
direct product of D12 and Z2 (also direct product of S3 and V4) 14 not nilpotent 2 8 4 0 12
direct product of E8 and Z3 15 1 1 24 0 0 24

Here is the GAP code to generate these data:

### Grouping by degrees of irreducible representations

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Total number of irreps = number of conjugacy classes Number of groups with these degrees of irreps Nilpotency class(es) attained Derived lengths attained Description of groups List of groups List of GAP IDs (second part)
24 0 0 24 3 1 1 all the abelian groups of order 24 cyclic group:Z24, direct product of Z6 and Z4, direct product of E8 and Z3 2, 9, 15
12 3 0 15 2 2 2 nilpotent non-abelian groups of order 24 direct product of D8 and Z3, direct product of Q8 and Z3 10, 11
8 4 0 12 4 non-nilpotent 2 nontrivial semidirect product of Z3 and Z8, direct product of S3 and Z4, direct product of Dic12 and Z2, direct product of D12 and Z2 (also described as direct product of S3 and V4) 1, 5, 7, 14
6 0 2 8 1 non-nilpotent 2 direct product of A4 and Z2 13
4 5 0 9 3 non-nilpotent 2 dicyclic group:Dic24, dihedral group:D24, SmallGroup(24,8) 4, 6, 8
3 3 1 7 1 non-nilpotent 3 special linear group:SL(2,3) 3
2 1 2 5 1 non-nilpotent 3 symmetric group:S4 12

### Correspondence between degrees of irreducible representations and conjugacy class sizes

For groups of order 24, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Number of conjugacy classes of size 1 Number of conjugacy classes of size 2 Number of conjugacy classes of size 3 Number of conjugacy classes of size 4 Number of conjugacy classes of size 6 Number of conjugacy classes of size 8 Total number of conjugacy classes = number of irreducible representations Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3
24 0 0 0 0 0 24 24 0 0
6 9 0 0 0 0 15 12 3 0
4 4 4 0 0 0 12 8 4 0
2 0 2 4 0 0 8 6 0 2
2 5 0 0 2 0 9 4 5 0
2 0 0 4 1 0 7 3 3 1
1 0 1 0 2 1 5 2 1 2

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders:

## Relation with other orders

### Divisors of the order

Divisor Quotient value Information on linear representation theory Relationship (subgroup perspective) Relationship (quotient perspective)
2 12 linear representation theory of cyclic group:Z2 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
3 8 linear representation theory of cyclic group:Z3 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
4 6 linear representation theory of groups of order 4 (linear representation theory of cyclic group:Z4, linear representation theory of Klein four-group) PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
6 4 linear representation theory of groups of order 6 (linear representation theory of cyclic group:Z6, linear representation theory of symmetric group:S3) PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
8 3 linear representation theory of groups of order 8 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
12 2 linear representation theory of groups of order 12 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

### Multiples of the order

Multiplier (other factor) Multiple Information on linear representation theory Relationship (subgroup perspective) Relationship (quotient perspective)
2 48 linear representation theory of groups of order 48
3 72 linear representation theory of groups of order 72
4 96 linear representation theory of groups of order 96
5 120 linear representation theory of groups of order 120
6 144 linear representation theory of groups of order 144
7 168 linear representation theory of groups of order 168