Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn(G) and dccn(G), as the smallest positive integer n such that C~n = G' for all non-central conjugacy classes C of G and Irr((χ~n)_G') = Irr(G') for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.
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