# Linear representation theory of groups of order 36

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 36.
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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

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Number of irreps of degree 4 Total number of irreps = number of conjugacy classes Number of groups with these degrees of irreps Nilpotency class(es) attained Derived lengths attained Description of groups List of groups List of GAP IDs (second part)
36 0 0 0 36 4 1 1 abelian groups 2, 5, 8, 14
12 6 0 0 18 2 not nilpotent 2 6, 12
9 0 3 0 12 2 not nilpotent 2 3, 11
4 0 0 2 6 1 not nilpotent 2 9
4 8 0 0 12 4 not nilpotent 2 1, 4, 7, 13
4 4 0 1 9 1 not nilpotent 2 10