# Linear representation theory of groups of order 81

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## 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

It turns out that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, so there are only three possibilities for the degrees of irreducible representations for groups of order 81, based on whether the nilpotency class is 1, 2, or 3.

Group GAP ID second part Nilpotency class Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree $3$ 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