Linear representation theory of groups of order 48
From Groupprops
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 |
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 |