Conjugacy class size statistics need not determine degrees of irreducible representations

Statement

It is possible to have two finite groups $G_{1}$ and $G_{2}$ such that the conjugacy class size statistics of $G_{1}$ are the same as those of $G_{2}$ (i.e., the two groups have the same number of conjugacy classes of each size) but the Degrees of irreducible representations (?) over $\mathbb {C}$ for $G_{1}$ are not the same as those of $G_{2}$.

Related facts

Converse

• Degrees of irreducible representations need not determine conjugacy class size statistics

Similar facts

• Conjugacy class size statistics need not determine group up to isoclinism
• Conjugacy class size statistics need not determine nilpotency class
• Conjugacy class size statistics need not determine derived length

Proof

Further information: Linear representation theory of groups of order 128#Degrees of irreducible representations, element structure of groups of order 128#Conjugacy class sizes

The smallest examples are among groups of order 128.