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

## Statement

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

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

## Related facts

For more facts, see degrees of irreducible representations.

### Related facts about conjugacy class sizes

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

## Proof

### Example of symmetric group of degree three

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 $p$ be a prime number. Consider an extraspecial group of order $p^5$ for any prime number $p$. The derived subgroup has order $p$. However, this group has a faithful irreducible representation of degree $p^2$.