Linear representation theory of groups of order 96
From Groupprops
This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 96.
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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
There are 29 possibilities for the degrees of irreducible representations for the 231 groups of order 96.
Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 | Number of irreps of degree 4 | Number of irreps of degree 6 | 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 GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|
96 | 0 | 0 | 0 | 0 | 96 | 7 | 1 | 1 | the abelian groups | 2, 46, 59, 161, 176, 220, 231 |
48 | 12 | 0 | 0 | 0 | 60 | 15 | 2 | 2 | ? | [SHOW MORE] |
48 | 0 | 0 | 3 | 0 | 51 | 2 | 2 | 2 | ? | 224, 225 |
32 | 16 | 0 | 0 | 0 | 48 | 12 | not nilpotent | 2 | ? | [SHOW MORE] |
24 | 18 | 0 | 0 | 0 | 42 | 19 | 2, 3 | 2 | ? | [SHOW MORE] |
24 | 6 | 0 | 3 | 0 | 33 | 5 | 3 | 2 | ? | 49, 50, 51, 183, 184 |
24 | 0 | 8 | 0 | 0 | 32 | 3 | not nilpotent | 2 | ? | 73, 196, 228 |
16 | 20 | 0 | 0 | 0 | 36 | 25 | not nilpotent | 2 | ? | [SHOW MORE] |
16 | 12 | 0 | 2 | 0 | 30 | 17 | not nilpotent | 2 | ? | [SHOW MORE] |
16 | 8 | 0 | 3 | 0 | 27 | 4 | not nilpotent | 2 | ? | 211, 214, 216, 217 |
12 | 21 | 0 | 0 | 0 | 33 | 3 | 4 | 2 | ? | 61, 62, 63 |
12 | 12 | 4 | 0 | 0 | 28 | 4 | not nilpotent | 3 | ? | 69, 74, 198, 200 |
12 | 3 | 4 | 0 | 1 | 20 | 2 | not nilpotent | 2 | ? | 197, 199 |
12 | 0 | 4 | 3 | 0 | 19 | 2 | not nilpotent | 3 | ? | 201, 202 |
8 | 22 | 0 | 0 | 0 | 30 | 19 | not nilpotent | 2 | ? | [SHOW MORE] |
8 | 14 | 0 | 2 | 0 | 24 | 38 | not nilpotent | 2 | ? | [SHOW MORE] |
8 | 10 | 0 | 3 | 0 | 21 | 18 | not nilpotent | 2 | ? | [SHOW MORE] |
8 | 6 | 0 | 4 | 0 | 18 | 4 | not nilpotent | 2 | ? | 118, 121, 122, 125 |
8 | 4 | 8 | 0 | 0 | 20 | 4 | not nilpotent | 3 | ? | 65, 186, 194, 226 |
6 | 0 | 10 | 0 | 0 | 16 | 2 | not nilpotent | 2 | ? | 68, 229 |
6 | 0 | 2 | 0 | 2 | 10 | 3 | not nilpotent | 2 | ? | 70, 71, 72 |
4 | 23 | 0 | 0 | 0 | 27 | 3 | not nilpotent | 2 | ? | 6, 7, 8 |
4 | 11 | 0 | 3 | 0 | 18 | 4 | not nilpotent | 2 | ? | 33, 34, 35, 36 |
4 | 6 | 4 | 2 | 0 | 16 | 5 | not nilpotent | 4 | ? | 66, 67, 188, 189, 192 |
4 | 5 | 4 | 0 | 1 | 14 | 3 | not nilpotent | 3 | ? | 185, 187, 195 |
4 | 2 | 4 | 3 | 0 | 13 | 3 | not nilpotent | 4 | ? | 190, 191, 193 |
3 | 3 | 5 | 0 | 1 | 12 | 2 | not nilpotent | 3 | ? | 3, 203 |
3 | 0 | 5 | 3 | 0 | 11 | 1 | not nilpotent | 3 | ? | 204 |
2 | 1 | 6 | 0 | 1 | 10 | 2 | not nilpotent | 3 | ? | 64, 227 |