I've just started studying representation theory of finite groups and I'm having trouble finding the character table of the group $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$.
This is what I've got so far (not sure if it's right):
$\begin{array}{c|c c c c c} \text{Class length} & 1 & 3 & 3 & 7 & 7 \\ \hline \text{Class rep.} & e & x & x^3 & y & y^2\\ \hline\hline 1&1&1&1&1&1\\ \tau_1 & 1 & 1 & 1 & \dfrac{1}{2}(-1+\sqrt{3} i) & \dfrac{1}{2}(-1-\sqrt{3} i)\\ \tau_2 & 1 & 1 & 1 & \dfrac{1}{2}(-1-\sqrt{3} i) & \dfrac{1}{2}(-1+\sqrt{3} i)\\ \tau_3 & 3\\ \tau_4 & 3\\ \end{array}$
But I don't know how to find the 3 dimensional ones. Could anyone offer help?