# Linear representation theory of groups of order 64

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 64.
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 64
To understand these in a broader context, see: linear representation theory of groups of order 2^n|linear representation theory of groups of prime-sixth order

## 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 64 mod 3, and hence congruent to 1 mod 3.

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 4 Total number of irreps (= number of conjugacy classes) Total number of groups Nilpotency class(es) attained by these Hall-Senior family/families List of GAP IDs (second part)
64 0 0 64 11 1 $\Gamma_1$, all the abelian groups of order 64 [SHOW MORE]
32 8 0 40 31 2 $\Gamma_2$ [SHOW MORE]
32 0 2 34 7 2 $\Gamma_5$ [SHOW MORE]
16 12 0 28 60 2,3 $\Gamma_3$ (class three), $\Gamma_4$ (class two) [SHOW MORE]
16 4 2 22 41 2,3 $\Gamma_6, \Gamma_7, ?$ [SHOW MORE]
8 14 0 22 33 2,3,4  ?, $\Gamma_8$ [SHOW MORE]