Abstract
We consider the problem of counting simultaneous conjugacy classes of n-tuples in a finite group G. Let denote the number of simultaneous conjugacy classes of n-tuples, and the number of simultaneous conjugacy classes of commuting n-tuples in G. We study the generating functions and which are rational functions of t. We show that determines and is completely determined by the class equation of G. We prove that the normalized functions and are invariants of isoclinism families. We compute exponential growth factors of and
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