Maximum degree of irreducible representation does not give bound on maximum conjugacy class size
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 versionPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- 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.
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 .