  Robinson showed that the character degrees are determined by knowing, for all
$n$, the number of ways that the identity can be expressed as a product of $n$
commutators. Earlier, Strunkov showed that the existence of characters of
$p$-defect 0 can be determined by counting solutions to certain equations
involving commutators and conjugates. In this paper, we prove analogs to
Robinson's and Strunkov's theorems by switching conjugacy classes and
characters. We show that counting the multiplicity of the trivial character in
certain products of characters determines the conjugacy class sizes and
existence of conjugacy classes with $p$-defect 0.
