Let $G$ be the unique non-abelian group of order 21 up to isomorphism. Compute the character table of $G$.

[You may find it helpful to think of $G$ as the group of $2 \times 2$ matrices of the form $\left(\begin{array}{cc}a & b \\ 0 & a^{-1}\end{array}\right)$ with $a, b \in \mathbb{F}_{7}$ and $a^{3}=1$. You may use any standard results from the course provided you state them clearly.]