Degrees of irreducible representations need not determine conjugacy class size statistics

Statement
It is possible to have two finite groups $$G_1$$ and $$G_2$$ such that the multiset of fact about::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.

Converse

 * Conjugacy class size statistics need not determine degrees of irreducible representations

Similar facts

 * Degrees of irreducible representations need not determine group up to isoclinism
 * Degrees of irreducible representations need not determine nilpotency class
 * Degrees of irreducible representations need not determine derived length

Proof
The smallest example orders are:


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