# Linear representation theory of special affine group:SA(2,3)

This article gives specific information, namely, linear representation theory, about a particular group, namely: special affine group:SA(2,3).
View linear representation theory of particular groups | View other specific information about special affine group:SA(2,3)

## Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 1,1,1,2,2,2,3,8,8,8
grouped form: 1 (3 times), 2 (3 times), 3 (1 time), 8 (3 times)
number: 10, maximum: 8, lcm: 24, sum of squares: 216

## 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

### Interpretation as special affine group of degree two over a finite field

Compare and contrast with linear representation theory of special affine group of degree two over a finite field

## GAP implementation

### Degrees of irreducible representations

The degrees of irreducible representations can be determined using GAP's CharacterDegrees and SA functions (note the SA function is not in-built):

gap> CharacterDegrees(SA(2,3));
[ [ 1, 3 ], [ 2, 3 ], [ 3, 1 ], [ 8, 3 ] ]