# Linear representation theory of groups of order 81

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 81.
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 81
To understand these in a broader context, see linear representation theory of groups of prime-fourth order | linear representation theory of groups of order 3^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
Group GAP ID second part Nilpotency class Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree $p$ Total number of irreps (= number of conjugacy classes)
Cyclic group:Z81 1 1 1 (81 times) 81 0 81
Direct product of Z9 and Z9 2 1 1 (81 times) 81 0 81
SmallGroup(81,3) 3 2 1 (27 times), 3 (6 times) 27 6 33
Nontrivial semidirect product of Z9 and Z9 4 2 1 (27 times), 3 (6 times) 27 6 33
Direct product of Z27 and Z3 5 1 1 (81 times) 81 0 81
Semidirect product of Z27 and Z3 6 2 1 (27 times), 3 (6 times) 27 6 33
Wreath product of Z3 and Z3 7 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,8) 8 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,9) 9 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,10) 10 3 1 (9 times), 3 (8 times) 9 8 17
Direct product of Z9 and E9 11 1 1 (81 times) 81 0 81
Direct product of prime-cube order group:U(3,3) and Z3 12 2 1 (27 times), 3 (6 times) 27 6 33
Direct product of M27 and Z3 13 2 1 (27 times), 3 (6 times) 27 6 33
Central product of prime-cube order group:U(3,3) and Z9 14 2 1 (27 times), 3 (6 times) 27 6 33
Elementary abelian group:E81 15 1 1 (81 times) 81 0 81