# Maximum degree of irreducible representation does not give bound on maximum conjugacy class size

## Contents

## Statement

### For a prime number

Suppose is a prime number. Then, for any positive integer , it is possible to construct a finite p-group such that the maximum degree of irreducible representation for is but has a conjugacy class of size .

### More general version

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## Related facts

- Maximum conjugacy class size does not give bound on maximum degree of irreducible representation
- Degrees of irreducible representations need not determine conjugacy class size statistics
- Conjugacy class size statistics need not determine degrees of irreducible representations

For more related facts, see the facts section of the degrees of irreducible representations page.

## Proof

### Proof case

`Further information: element structure of dihedral groups, linear representation theory of dihedral groups`

In this case, we can take to be any of the three maximal class groups of order (see classification of finite 2-groups of maximal class). For instance, we could take the dihedral group of order and degree .

As per the linear representation theory of dihedral groups, all the degrees of irreducible representations are either 1 or 2, whereas the largest conjugacy class size is .

### Proof for odd

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