Quasirandom degree of extension group is bounded below by minimum of quasirandom degrees of normal subgroup and quotient group

In terms of quasirandom degree
Suppose $$G$$ is a finite group and $$N$$ is a normal subgroup of $$G$$. Then, the quasirandom degree of $$G$$ is at least equal to the minimum of the quasirandom degrees of $$N$$ and of the quotient group $$G/N$$.

The quasirandom degree is the minimum possible degree of a nontrivial linear representation over the field of complex numbers.

In terms of $$D$$-quasirandomness
Suppose $$G$$ is a finite group and $$N$$ is a normal subgroup of $$G$$. If $$D$$ is a positive integer such that both $$N$$ and $$G/N$$ are $$D$$-quasirandom, then $$G$$ is also $$D$$-quasirandom.

Related facts

 * Quasirandom degree of quotient group is bounded below by quasirandom degree of whole group
 * Quasirandom degree of group is bounded below by minimum of quasirandom degrees of generating subgroups

Proof
It suffices to show that if $$G$$ has a nontrivial linear representation of degree $$d$$ over the field of complex numbers, then either $$N$$ or $$G/N$$ has a nontrivial linear representation of degree $$d$$.

Given: A finite group $$G$$, normal subgroup $$N$$. A nontrivial linear representation $$\varphi$$ of $$G$$ of degree $$d$$.

To prove: Either $$N$$ or $$G/N$$ has a nontrivial linear representation of degree $$d$$.

Proof: Consider the restriction of $$\varphi$$ to $$N$$. This is a linear representation of $$N$$ of degree $$d$$. If this is nontrivial, we are done. If it is trivial, $$\varphi$$ descends to a linear representation of $$G/N$$ of the same degree $$d$$, which must be nontrivial because $$\varphi$$ itself is nontrivial. In that case, we are done too.