# Linear representation theory of groups of order 27

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## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 27.
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 27
To understand these in a broader context, see linear representation theory of groups of prime-cube order |linear representation theory of groups of order 3^n
Group GAP ID 2nd part Linear representation theory page
cyclic group:Z27 1 linear representation theory of cyclic group:Z27
direct product of Z9 and Z3 2 linear representation of direct product of Z9 and Z3
prime-cube order group:U(3,3) 3 linear representation theory of prime-cube order group:U(3,3)
M27 (semidirect product of Z9 and Z3) 4 linear representation theory of M27
elementary abelian group:E27 5 linear representation theory of elementary abelian group:E27

## 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 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:Z27 1 1 (27 times) 27 0 27
direct product of Z9 and Z3 2 1 (27 times) 27 0 27
prime-cube order group:U(3,3) 3 1 (9 times), 3 (2 times) 9 2 11
M27 4 1 (9 times), 3 (2 times) 9 2 11
elementary abelian group:E27 5 1 (27 times) 27 0 27