A question about relation of the character degrees of $G/N$ and $G$

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.

• Could you add some motivation for your question? – Mark Wildon Feb 11 '17 at 14:41
• 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. – BHZ Feb 11 '17 at 18:23

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 $\{1,2,3\}$.