Linear representation theory of groups of order 81
From Groupprops
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  |
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 |
