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

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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 but G has a conjugacy class of size p^m.

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

Proof case p = 2

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

In this case, we can take G to be any of the three maximal class groups of order 2^{m+2} (see classification of finite 2-groups of maximal class). For instance, we could take the dihedral group of order 2^{m+2} and degree 2^{m+1}.

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 2^m.

Proof for odd p

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