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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?

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    There are a lot of ways to proceed depending on what tools you have at your disposal. You can aggressively exploit orthogonality and the fact that you know the character of the regular representation. You can try to write down induced representations. You can try to write down permutation representations...2012-03-09
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    It seems reasonable to try those induced characters of linear ones of a normal subgroup of index 3, which this group must possess. I have not yet worked out the details, but if they turn out to be the remaining irreducible characters, then they are all monomial and hence G is a M-group. Well, all speculations.2012-05-23

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