Linear representation theory of groups of order 128

This article describes the linear representation theory of groups of order 128. There are a total of 2328 groups of order 128, so we do not present information group by group but rather present key summary information.

Grouping by degrees of irreducible representations
Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the total number of irreducible representations, which equals the number of conjugacy classes, is congruent to 128 mod 3, and hence congruent to 2 mod 3.

Here is GAP code to generate this information:

We use the coded function IrrepDegreeGroupingFull (not in-built):

gap> I := IrrepDegreeGroupingFull(128);; gap> List(I,x -> [x[1],Length(x[2]),Set(List(x[2],i -> NilpotencyClassOfGroup(SmallGroup(128,i)))),Set(List(x[2],i->DerivedLength(SmallGroup(128,i))))]); [ [ [ [ 1, 4 ], [ 2, 31 ] ], 3, [ 6 ], [ 2 ] ], [ [ [ 1, 8 ], [ 2, 2 ], [ 4, 3 ], [ 8, 1 ] ], 4, [ 5 ], [ 2 ] ], [ [ [ 1, 8 ], [ 2, 2 ], [ 4, 7 ] ], 9, [ 5 ], [ 2, 3 ] ], [ [ [ 1, 8 ], [ 2, 6 ], [ 4, 2 ], [ 8, 1 ] ], 5, [ 4 ], [ 3 ] ],  [ [ [ 1, 8 ], [ 2, 6 ], [ 4, 6 ] ], 15, [ 3, 4 ], [ 2, 3 ] ], [ [ [ 1, 8 ], [ 2, 10 ], [ 4, 1 ], [ 8, 1 ] ], 10, [ 4 ], [ 2 ] ],  [ [ [ 1, 8 ], [ 2, 10 ], [ 4, 5 ] ], 65, [ 3, 4 ], [ 2 ] ], [ [ [ 1, 8 ], [ 2, 14 ], [ 4, 4 ] ], 108, [ 3, 4, 5 ], [ 2 ] ],  [ [ [ 1, 8 ], [ 2, 14 ], [ 8, 1 ] ], 10, [ 3 ], [ 2 ] ], [ [ [ 1, 8 ], [ 2, 18 ], [ 4, 3 ] ], 54, [ 3, 4 ], [ 2 ] ],  [ [ [ 1, 8 ], [ 2, 22 ], [ 4, 2 ] ], 52, [ 3 ], [ 2 ] ], [ [ [ 1, 8 ], [ 2, 30 ] ], 32, [ 3, 4, 5 ], [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 4 ], [ 4, 2 ], [ 8, 1 ] ], 11, [ 4 ], [ 2 ] ], [ [ [ 1, 16 ], [ 2, 4 ], [ 4, 6 ] ], 134, [ 2, 3, 4 ], [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 8 ], [ 4, 1 ], [ 8, 1 ] ], 2, [ 3 ], [ 2 ] ], [ [ [ 1, 16 ], [ 2, 8 ], [ 4, 5 ] ], 88, [ 3 ], [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 12 ], [ 4, 4 ] ], 538, [ 2, 3, 4 ], [ 2 ] ], [ [ [ 1, 16 ], [ 2, 12 ], [ 8, 1 ] ], 25, [ 3 ], [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 16 ], [ 4, 3 ] ], 50, [ 3 ], [ 2 ] ], [ [ [ 1, 16 ], [ 2, 20 ], [ 4, 2 ] ], 361, [ 2, 3 ], [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 28 ] ], 149, [ 2, 3, 4 ], [ 2 ] ], [ [ [ 1, 32 ], [ 2, 8 ], [ 4, 4 ] ], 204, [ 2, 3 ], [ 2 ] ],  [ [ [ 1, 32 ], [ 2, 8 ], [ 8, 1 ] ], 6, [ 3 ], [ 2 ] ], [ [ [ 1, 32 ], [ 2, 16 ], [ 4, 2 ] ], 80, [ 2 ], [ 2 ] ],  [ [ [ 1, 32 ], [ 2, 24 ] ], 158, [ 2, 3 ], [ 2 ] ], [ [ [ 1, 32 ], [ 4, 6 ] ], 57, [ 2 ], [ 2 ] ], [ [ [ 1, 64 ], [ 2, 16 ] ], 60, [ 2 ], [ 2 ] ],  [ [ [ 1, 64 ], [ 4, 4 ] ], 21, [ 2 ], [ 2 ] ], [ [ [ 1, 64 ], [ 8, 1 ] ], 2, [ 2 ], [ 2 ] ], [ [ [ 1, 128 ] ], 15, [ 1 ], [ 1 ] ] ]