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

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Statement

For a prime number

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

More general version

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Related facts

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

Proof

Further information: element structure of extraspecial groups, linear representation theory of extraspecial groups

Take the extraspecial group of order p^{2m +  1} (there are two such groups and either will do). The maximum degree of irreducible representation for this group is p^m, and the maximum conjugacy class size is p.