Character table guarantees certain property of groups

Question

How to show there exist three conjugacy classes in a certain group $G$ such that the products $xyz, x\in C_1, y\in C_2, z\in C_3$ divide equally over all elements of $G$ by looking at the character table?

For example, take $G=A_5$, $x\in [(12)(34)]=C_1,y\in [(123)]=C_2,z\in [(12345)]=C_3$, then each element $a\in A_5$ can be written in the same number of ways as $a=xyz$. How to deduce this by means of properties of the character table?

Thanks.

Answer 1

This happens if and only if $$\sum_{\chi\in\mathrm{Irr}(G)}\frac{\chi(x)\chi(y)\chi(z)\chi(g)}{\chi(1)^2}$$ is constant (i.e. does not depend on $g$). See Corollary 4.14 in [Navarro, Characer theory and the McKay Conjecture, CUP, 2018].