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 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

• 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 $243=3^{5}$