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Let $ G $ be a finite group and $ N $ be a normal subgroup of $ G $. Let $ G/N $ have two irreducible characters of degrees $ p_1$ and $ p_2$, where $ p_1$ and $ p_2$ are different primes. Let $ G/N $ have no irreducible character such that $ p_1p_2\mid \chi (1) $. If $ (p_1p_2, |N|)=1$, can we say that $G $ has no irreducible character $\chi $ such that $ p_1p_2\mid \chi (1) $? I guess that this is impossible but I could not find any counterexample for it.

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  • $\begingroup$ Could you add some motivation for your question? $\endgroup$ – Mark Wildon Feb 11 '17 at 14:41
  • $\begingroup$ Thank you very much for your very nice example. This topic is related to the structure of character degree graph of a finite group and I want to know the relations between the character degree graphs. $\endgroup$ – BHZ Feb 11 '17 at 18:23
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It is possible for $G$ to have such a character. The smallest example has order $120$. Let $N = \langle k \rangle \cong C_5$. Let $G = N \rtimes S_4$ where the action of the symmetric group $S_4$ on $N$ is non-trivial but factors through the sign representation. Thus $k^{(12)} = k^{-1}$, $k^{(234)} = k$ and $\langle k, (12)\rangle$ is dihedral of order $10$. The character degrees of $G$ are $\{1,2,3,6\}$ and the character degrees of $G/N \cong S_4$ are $\{1,2,3\}$.

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