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Suppose G is a semi-direct product of P with H where P is a (non-abelian) p-group and G is solvable. I wonder what can be said about the irreducible characters of G given information about the characters of P and H.

In particular, I want to know if one can say anything about the monomial irreducible characters of G given we know all monomial irreducible characters of P and H (every irr. character of P is monomial since P is nilpotent).

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