 Solution

For a representation $\rho$ of $G/N$, its inflation of a group representation to $G$ is $g\mapsto \rho (gN)$, with character $\tilde{\chi }(g)=\chi (gN)$. Inflation preserves irreducibility because the invariant subspaces do not change.

For irreducible complex characters ${\chi }_i$, the row character orthogonality areThe column relations are ${\sum }_i{\chi }_i(g)\overline{{\chi }_i(h)}=0$ for nonconjugate $g,h$, and $|C_G(g)|$ when $g,h$ are conjugate. Class sizes convert the row sum into its usual weighted character-table form.