# Degree of irreducible representation need not divide order of derived subgroup

## Contents

## Statement

It is possible to have a finite group and an irreducible linear representation of over a splitting field in characteristic zero such that the degree of does not divide the order of the Derived subgroup (?) of .

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

## Related facts

For more facts, see degrees of irreducible representations.

### Similar facts

- Degree of irreducible representation may be greater than order of derived subgroup
- Degree of irreducible representation need not divide exponent
- Degree of irreducible representation may be greater than exponent

### Opposite facts

- Degree of irreducible representation divides order of group
- Degree of irreducible representation divides index of center
- Degree of irreducible representation divides index of abelian normal subgroup
- Degree of irreducible representation is bounded by index of abelian subgroup

### Related facts about conjugacy class sizes

These facts are related to the conjugacy class size statistics of a finite group.

- Size of conjugacy class is bounded by order of derived subgroup
- Size of conjugacy class need not divide order of derived subgroup

## Proof

### Example of symmetric group of degree three

`Further information: symmetric group:S3, subgroup structure of symmetric group:S3, linear representation theory of symmetric group:S3`

We consider the group symmetric group:S3. The standard representation of symmetric group:S3 is a faithful irreducible representation of degree two, and the derived subgroup, A3 in S3, has order three.

### Example of extraspecial group

Let be a prime number. Consider an extraspecial group of order for any prime number . The derived subgroup has order . However, this group has a faithful irreducible representation of degree .