I read character theory recently and thought about the following proposition, but I do not know is this true or false:
Let $G $ be a finite group such that two distinct primes $ p $ and $ q $ divide the order of $ G $. Also $ G $ has a normal subgroup $ N $ such that $|N|$ is coprime to $ pq $, i.e. $(|N|,pq)= 1$. If $ G/N $ has some irreducible characters of degree $ p $ and $ q $, but has no irreducible character of degree $ pq $, is it true that $ G$ has no irreducible character of degree divisible by $ pq $?
Does anyone have any insight?