Character-conjugate conjugacy classes

Definition
Two conjugacy classes in a group are said to be character-conjugate if for every finite-dimensional indecomposable linear representation (and hence any finite-dimensional linear representation) of the group, the value that the character of the representation takes on both the conjugacy classes is equal.

Stronger properties

 * Locally conjugate conjugacy classes

Facts
It turns out that for a finite group, and a field whose characteristic does not divide the order of the group, character-conjugate conjugacy classes are the same as locally conjugate conjugacy classes.