# Linear representation theory of groups of order 128

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 128.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about 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.

To understand these in a broader context, see linear representation theory of groups of prime-seventh order |linear representation theory of groups of order 2^n

## 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

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.

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 4 Number of irreps of degree 8 Total number of irreps (= number of conjugacy classes) Total number of groups Nilpotency class(es) attained by these Derived length(s) attained by these Description of groups
128 0 0 0 128 15 1 1 abelian groups
64 16 0 0 80 60 2 2
64 0 4 0 68 21 2 2
64 0 0 1 65 2 2 2 extraspecial groups
32 24 0 0 56 158 2,3 2
32 0 6 0 38 57 2 2
32 16 2 0 50 80 2 2
32 8 4 0 44 204 2,3 2
32 8 0 1 41 6 3 2
16 28 0 0 44 149 2,3,4 2
16 20 2 0 38 361 2,3 2
16 16 3 0 35 50 3 2
16 12 4 0 32 538 2,3,4 2
16 12 0 1 29 25 3 2
16 8 5 0 29 88 3 2
16 8 1 1 26 2 3 2
16 4 6 0 26 134 2,3,4 2
16 4 2 1 21 11 4 2
8 30 0 0 38 32 3,4,5 2
8 22 2 0 32 52 3 2
8 18 3 0 29 54 3,4 2
8 14 0 1 23 10 3 2
8 14 4 0 26 108 3,4,5 2
8 10 5 0 23 65 3,4 2
8 10 1 1 20 10 4 2
8 6 6 0 20 15 3,4 2,3
8 6 2 1 17 5 4 3
8 2 7 0 17 9 5 2,3
8 2 3 1 14 4 5 2
4 31 0 0 35 3 6 2 the maximal class groups