= Solution
For a representation $\rho$ of $G/N$, its <inflation of a group representation> to $G$ is $g\mapsto\rho(gN)$, with character $\widetilde\chi(g)=\chi(gN)$. Inflation preserves irreducibility because the invariant subspaces do not change.
For irreducible complex characters $\chi_i$, the row <character orthogonality> are
$$
\frac1{|G|}\sum_{g\in G}\chi_i(g)\overline{\chi_j(g)}=\delta_{ij}.
$$
The 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.
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