A finite group $G$ of order 360 has conjugacy classes $C_{1}=\{1\}, C_{2}, \ldots, C_{7}$ of sizes $1,45,40,40,90,72,72$. The values of four of its irreducible characters are given in the following table.

$$\begin{array}{ccccccc} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \\ 5 & 1 & 2 & -1 & -1 & 0 & 0 \\ 8 & 0 & -1 & -1 & 0 & (1-\sqrt{5}) / 2 & (1+\sqrt{5}) / 2 \\ 8 & 0 & -1 & -1 & 0 & (1+\sqrt{5}) / 2 & (1-\sqrt{5}) / 2 \\ 10 & -2 & 1 & 1 & 0 & 0 & 0 \end{array}$$

Complete the character table.

[Hint: it will not suffice just to use orthogonality of characters.]

Deduce that the group $G$ is simple.