# Linear representation theory of groups of order 96

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