# Linear representation theory of groups of order 48

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 48.
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## 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

### Grouping by degrees of irreducible representations

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Number of irreps of degree 4 Total number of irreducible representations = number of conjugacy classes Number of groups with these irreps Nilpotency class(es) attained Derived lengths attained Description of groups List of groups List of GAP IDs (second part)
48 0 0 0 48 5 1 1 all the abelian groups 2, 20, 23, 44, 52
24 6 0 0 30 6 2 2 all the non-abelian groups of nilpotency class two, obtained as direct products of such groups of order 16 and cyclic group:Z3 21, 22, 24, 45, 46, 47
16 8 0 0 24 7 not nilpotent 2 1, 4, 9, 11, 35, 42, 51
12 9 0 0 21 3 3 2 all the groups of nilpotency class three, obtained as direct products of such groups of order 16 with cyclic group:Z3 direct product of D16 and Z3, direct product of SD16 and Z3, direct product of Q16 and Z3 25, 26, 27
12 0 4 0 16 2 not nilpotent 2 31, 49
8 10 0 0 18 10 not nilpotent 2 5, 10, 12, 13, 14, 19, 34, 36, 37, 43
8 6 0 1 15 4 not nilpotent 2 38, 39, 40, 41
6 6 2 0 14 2 not nilpotent 3 32, 33
4 11 0 0 15 3 not nilpotent 2 6, 7, 8
4 7 0 1 12 4 not nilpotent 2 15, 16, 17, 18
4 2 4 0 10 2 not nilpotent 3 30, 48
3 0 5 0 8 2 not nilpotent 2 3, 50
2 3 2 1 8 2 not nilpotent 4 all the derived length four groups general linear group:GL(2,3) and binary octahedral group 28, 29
The GAP code to construct this is as follows:[SHOW MORE]

## Relation with other orders

### Divisors of the order

Divisor Quotient value Information on linear representation theory Relationship (subgroup perspective) Relationship (quotient perspective)
2 24 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 16 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 12 linear representation theory of groups of order 4 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
6 8 linear representation theory of groups of order 6 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
8 6 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 4 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]
16 3 linear representation theory of groups of order 16 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
24 2 linear representation theory of groups of order 24 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 96 linear representation theory of groups of order 96
3 144 linear representation theory of groups of order 144
4 192 linear representation theory of groups of order 192
5 240 linear representation theory of groups of order 240
6 288 linear representation theory of groups of order 288
7 336 linear representation theory of groups of order 336
8 384 linear representation theory of groups of order 384
9 432 linear representation theory of groups of order 432
10 480 linear representation theory of groups of order 480