Do the class vector and character vector of a $p$-group determine each other? To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$:


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*The class vector - $v_i$ is the number of conjugacy classes of order $p^i$.

*The character vector - $v_i$ is the number of complex irreducible representations of dimension $i$ up to isomorphism.
Question: Do these two invariants of the group determine each other? In other words, if two groups have the same class vector, do they have the same character vector and vice versa?

 A: The answer is no. This observation is cataloged at the Group Properties Wiki on pages 1 and 2.

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 $G_2$...
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...

Further, the smallest example of the first statement is of size $128$, while for the other statement, examples of groups of size $64$ exist so that these statements are true for $p$-groups as well.
