A finite group $G$ has seven conjugacy classes $C_{1}=\{e\}, C_{2}, \ldots, C_{7}$ and the values of five of its irreducible characters are given in the following table.

$\begin{array}{rrrrrrr}C_{1} & C_{2} & C_{3} & C_{4} & C_{5} & C_{6} & C_{7} \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 \\ 4 & 0 & 1 & -1 & 2 & -1 & 0 \\ 4 & 0 & 1 & -1 & -2 & 1 & 0 \\ 5 & 1 & -1 & 0 & 1 & 1 & -1\end{array}$

Calculate the number of elements in the various conjugacy classes and complete the character table.

[You may not identify $G$ with any known group, unless you justify doing so.]