# Linear representation theory of groups of order 60

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 60.
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 60

## 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 Number of irreps of degree 5 Total number of irreps = number of conjugacy classes Number of groups with these irreps Nilpotency class(es) attained Derived length(s) attained Description of groups List of groups List of GAP IDs (second part)
60 0 0 0 0 60 2 1 1 all the abelian groups cyclic group:Z60, direct product of V4 and Z15 4, 13
20 10 0 0 0 30 2 not nilpotent 2  ?  ? 1, 11
15 0 5 0 0 20 1 not nilpotent 2  ?  ? 9
12 12 0 0 0 24 2 not nilpotent 2  ?  ? 2, 10
12 0 0 3 0 15 1 not nilpotent 2  ?  ? 6
4 14 0 0 0 18 2 not nilpotent 2  ?  ? 3, 12
4 6 0 2 0 12 1 not nilpotent 2  ?  ? 8
4 2 0 3 0 11 1 not nilpotent 2  ?  ? 7
1 0 2 1 1 5 1 not nilpotent not solvable The unique simple non-abelian group of this order alternating group:A5 5