I am trying to comple the character table of a finite group with seven conjugacy classes $c_1, \cdots, c_7$:
If I use the orthogonality of the column vectors of table I get a system of $21$ nonlinear equations of the form $$ k_{i, j} + r_i r_j + s_i s_j = 0 $$
where $k _{i, j} \in \mathbb{R}$ for $1 \leq i, j \leq 7$ and $i \neq j$. I do not know if it helps but I noticed that I can write evey equation as a $2 \times 2$ matrix with determinant $-k_{i, j}$: $$ \operatorname{det} \begin{pmatrix} r_i & -s_j\\ s_i & r_j \end{pmatrix} = -k_{i, j}. $$
I do not know how to solve this system. Can you help me?
If you multiply the fifth character with the second, you obtain another irreducible character. The remaining missing character should be easy to determine.