# Degree of irreducible representation need not divide exponent

## Contents

## Statement

We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero (or more generally over a splitting field), such that the degree of the irreducible representation does *not* divide the exponent of the group.

This is a non-constraint on the Degrees of irreducible representations (?) of a finite group.

## Related facts

### Similar facts

- Degree of irreducible representation may be greater than exponent
- Square of degree of irreducible representation need not divide order
- Size of conjugacy class need not divide exponent

### Opposite facts

- Schur index of irreducible character in characteristic zero divides exponent, Schur index divides degree of irreducible representation: Thus, the Schur index of an irreducible character/representation divides both the degree of the representation
*and*the exponent. - Degree of irreducible representation divides group order
- Degree of irreducible representation divides order of inner automorphism group (i.e., the degree divides the index of the center)
- Degree of irreducible representation divides index of abelian normal subgroup
- Order of inner automorphism group bounds square of degree of irreducible representation

## Proof

### Example of big extraspecial groups

Consider an extraspecial group of order for any prime . The exponent of this group is either or . On the other hand, it admits a faithful irreducible representation of degree .

For odd primes , we can take an example of an extraspecial group of order and exponent , which admits a faithful irreducible representation of degree .

### Example of symmetric group of degree six

`Further information: symmetric group:S6, linear representation theory of symmetric group:S6`

The symmetric group of degree six has an irreducible representation of degree over the rational numbers, arising from the self-conjugate partition . On the other hand, the exponent of the group is the lcm of the numbers from to , which is , and is not a multiple of .