Character-conjugate conjugacy classes
This term is related to: linear representation theory
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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.
Relation with other properties
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.