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] 45, 47, 48, 55, 60, 162, 163, 164, 165, 166, 177, 178, 221, 222, 223
48	0	0	3	0	51	2	2	2	?	224, 225
32	16	0	0	0	48	12	not nilpotent	2	?	[SHOW MORE] 1, 4, 9, 18, 20, 78, 106, 127, 129, 206, 218, 230
24	18	0	0	0	42	19	2, 3	2	?	[SHOW MORE] 52, 53, 54, 56, 57, 58, 167, 168, 169, 170, 171, 172, 173, 174, 175, 179, 180, 181, 182
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] 5, 10, 11, 19, 21, 22, 27, 37, 38, 75, 79, 80, 107, 108, 128, 130, 132, 133, 134, 135, 159, 205, 207, 208, 219
16	12	0	2	0	30	17	not nilpotent	2	?	[SHOW MORE] 84, 87, 88, 94, 98, 99, 100, 113, 114, 141, 152, 155, 209, 210, 212, 213, 215
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] 12, 23, 24, 25, 26, 28, 76, 77, 81, 82, 83, 109, 110, 111, 112, 131, 136, 137, 160
8	14	0	2	0	24	38	not nilpotent	2	?	[SHOW MORE] 14, 15, 16, 17, 29, 32, 39, 42, 44, 85, 86, 89, 90, 91, 92, 93, 95, 96, 97, 101, 102, 103, 104, 105, 138, 140, 142, 143, 144, 145, 146, 147, 148, 150, 151, 153, 154, 157
8	10	0	3	0	21	18	not nilpotent	2	?	[SHOW MORE] 13, 30, 31, 40, 41, 43, 115, 116, 117, 119, 120, 123, 124, 126, 139, 149, 156, 158
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

Here is the GAP code to generate this:[SHOW MORE]

We use the GAP function (not in-built, but coded) IrrepDegreeGroupingFull and ModifiedNilpClass (the latter simply returns the nilpotency class if the group is nilpotent and 0 otherwise):

gap> I := IrrepDegreeGroupingFull(96);;
gap> List(I,x -> [x[1],Length(x[2]),Set(List(x[2],i -> ModifiedNilpClass(SmallGroup(96,i)))),Set(List(x[2],i -> DerivedLength(SmallGroup(96,i)))),x[2]]);
[ [ [ [ 1, 2 ], [ 2, 1 ], [ 3, 6 ], [ 6, 1 ] ], 2, [ 0 ], [ 3 ], [ 64, 227 ] ], [ [ [ 1, 3 ], [ 2, 3 ], [ 3, 5 ], [ 6, 1 ] ], 2, [ 0 ], [ 3 ], [ 3, 203 ]
     ], [ [ [ 1, 3 ], [ 3, 5 ], [ 4, 3 ] ], 1, [ 0 ], [ 3 ], [ 204 ] ], [ [ [ 1, 4 ], [ 2, 2 ], [ 3, 4 ], [ 4, 3 ] ], 3, [ 0 ], [ 4 ], [ 190, 191, 193 ] ]
    , [ [ [ 1, 4 ], [ 2, 5 ], [ 3, 4 ], [ 6, 1 ] ], 3, [ 0 ], [ 3 ], [ 185, 187, 195 ] ],
  [ [ [ 1, 4 ], [ 2, 6 ], [ 3, 4 ], [ 4, 2 ] ], 5, [ 0 ], [ 4 ], [ 66, 67, 188, 189, 192 ] ],
  [ [ [ 1, 4 ], [ 2, 11 ], [ 4, 3 ] ], 4, [ 0 ], [ 2 ], [ 33, 34, 35, 36 ] ], [ [ [ 1, 4 ], [ 2, 23 ] ], 3, [ 0 ], [ 2 ], [ 6, 7, 8 ] ],
  [ [ [ 1, 6 ], [ 3, 2 ], [ 6, 2 ] ], 3, [ 0 ], [ 2 ], [ 70, 71, 72 ] ], [ [ [ 1, 6 ], [ 3, 10 ] ], 2, [ 0 ], [ 2 ], [ 68, 229 ] ],
  [ [ [ 1, 8 ], [ 2, 4 ], [ 3, 8 ] ], 4, [ 0 ], [ 3 ], [ 65, 186, 194, 226 ] ],
  [ [ [ 1, 8 ], [ 2, 6 ], [ 4, 4 ] ], 4, [ 0 ], [ 2 ], [ 118, 121, 122, 125 ] ],
  [ [ [ 1, 8 ], [ 2, 10 ], [ 4, 3 ] ], 18, [ 0 ], [ 2 ], [ 13, 30, 31, 40, 41, 43, 115, 116, 117, 119, 120, 123, 124, 126, 139, 149, 156, 158 ] ],
  [ [ [ 1, 8 ], [ 2, 14 ], [ 4, 2 ] ], 38, [ 0 ], [ 2 ], [ 14, 15, 16, 17, 29, 32, 39, 42, 44, 85, 86, 89, 90, 91, 92, 93, 95, 96, 97, 101, 102, 103,
          104, 105, 138, 140, 142, 143, 144, 145, 146, 147, 148, 150, 151, 153, 154, 157 ] ],
  [ [ [ 1, 8 ], [ 2, 22 ] ], 19, [ 0 ], [ 2 ], [ 12, 23, 24, 25, 26, 28, 76, 77, 81, 82, 83, 109, 110, 111, 112, 131, 136, 137, 160 ] ],
  [ [ [ 1, 12 ], [ 2, 3 ], [ 3, 4 ], [ 6, 1 ] ], 2, [ 0 ], [ 2 ], [ 197, 199 ] ],
  [ [ [ 1, 12 ], [ 2, 12 ], [ 3, 4 ] ], 4, [ 0 ], [ 3 ], [ 69, 74, 198, 200 ] ], [ [ [ 1, 12 ], [ 2, 21 ] ], 3, [ 4 ], [ 2 ], [ 61, 62, 63 ] ],
  [ [ [ 1, 12 ], [ 3, 4 ], [ 4, 3 ] ], 2, [ 0 ], [ 3 ], [ 201, 202 ] ], [ [ [ 1, 16 ], [ 2, 8 ], [ 4, 3 ] ], 4, [ 0 ], [ 2 ], [ 211, 214, 216, 217 ] ],
  [ [ [ 1, 16 ], [ 2, 12 ], [ 4, 2 ] ], 17, [ 0 ], [ 2 ], [ 84, 87, 88, 94, 98, 99, 100, 113, 114, 141, 152, 155, 209, 210, 212, 213, 215 ] ],
  [ [ [ 1, 16 ], [ 2, 20 ] ], 25, [ 0 ], [ 2 ], [ 5, 10, 11, 19, 21, 22, 27, 37, 38, 75, 79, 80, 107, 108, 128, 130, 132, 133, 134, 135, 159, 205, 207,
          208, 219 ] ], [ [ [ 1, 24 ], [ 2, 6 ], [ 4, 3 ] ], 5, [ 3 ], [ 2 ], [ 49, 50, 51, 183, 184 ] ],
  [ [ [ 1, 24 ], [ 2, 18 ] ], 19, [ 2, 3 ], [ 2 ], [ 52, 53, 54, 56, 57, 58, 167, 168, 169, 170, 171, 172, 173, 174, 175, 179, 180, 181, 182 ] ],
  [ [ [ 1, 24 ], [ 3, 8 ] ], 3, [ 0 ], [ 2 ], [ 73, 196, 228 ] ],
  [ [ [ 1, 32 ], [ 2, 16 ] ], 12, [ 0 ], [ 2 ], [ 1, 4, 9, 18, 20, 78, 106, 127, 129, 206, 218, 230 ] ],
  [ [ [ 1, 48 ], [ 2, 12 ] ], 15, [ 2 ], [ 2 ], [ 45, 47, 48, 55, 60, 162, 163, 164, 165, 166, 177, 178, 221, 222, 223 ] ],
  [ [ [ 1, 48 ], [ 4, 3 ] ], 2, [ 2 ], [ 2 ], [ 224, 225 ] ], [ [ [ 1, 96 ] ], 7, [ 1 ], [ 1 ], [ 2, 46, 59, 161, 176, 220, 231 ] ] ]