Degrees of irreducible representations need not determine conjugacy class size statistics

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Statement

It is possible to have two finite groups G_1 and G_2 such that the multiset of Degrees of irreducible representations (?) (over \mathbb{C}) of G_1 is the same as the multiset of degrees of irreducible representations of G_2 (i.e., G_1 and G_2 have the same number of irreducible representations of each degree) but the conjugacy class size statistics of G_1 and G_2 are not the same.

Related facts

Converse

Similar facts

Proof

The smallest example orders are:

  • Order 64 = 2^6
  • Order 96 = 2^5 \cdot 3
  • Order 243 = 3^5