Conjugacy class size statistics need not determine degrees of irreducible representations

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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

Similar facts

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.