Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field whose characteristic does not divide the group order.

1) Let $\operatorname{Ind}_S^G(f)$ have the degree $n$ and decompose into irreducible representations $f_1,\dots, f_k$ of $G$. Is there a formula like $$n=\deg(f_1 \oplus \cdots \oplus f_k)=\cdots?$$

2) And a general question: If I have the number of irreducible representations and their degrees, is there a way to find out how many of them are faithful?

Thank you very much.


1) Yes, it is the sum of the degrees of the subrepresentation.

2) No.


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