To a finite $p$-group, we can associate two vectors $(v\_0,v\_1,\dotsc)$:


1. The class vector - $v\_i$ is the number of conjugacy classes of order $p^i$.
2. 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?




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Do the class vector and character vector of a $p$-group determine each other?
The answer is no. This observation is cataloged at the Group Properties Wiki on pages [1](https://groupprops.subwiki.org/wiki/Conjugacy_class_size_statistics_need_not_determine_degrees_of_irreducible_representations) and [2](https://groupprops.subwiki.org/wiki/Degrees_of_irreducible_representations_need_not_determine_conjugacy_class_size_statistics).



> 
> 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.


