Conjugacy class of more than average size has character value zero for some irreducible character

Statement
Suppose $$G$$ is a finite group and $$c$$ is a conjugacy class in $$G$$ such that $$c$$ has size more than the average conjugacy class size, i.e., $$|G|/|c|$$ is less than the number of conjugacy classes in $$G$$.

Then, there exists an irreducible representation of $$G$$ over the complex numbers with character $$\chi$$ such that $$\chi(g) = 0$$ for some $$g \in c$$.

Related facts

 * Central implies image under every irreducible representation is scalar
 * Irreducible character of degree greater than one takes value zero on some conjugacy class
 * Zero-or-scalar lemma