I am trying to find
The number of nonisomoprhic simple modules of $\mathbb{C}G$, where $G = \langle s,t \mid s^8=t^2=1, st=ts^3 \rangle$.
This is equivalent to (I think, by Artin-Wedderburn) the number of conjugacy classes of $G$. We may represent any element in $G$ in the form $s^kt^l$ for $1 \le k \le 8, 1 \le l \le 2$, using the relations. So
How does one compute the number of conjugacy classes of $G$? Any hints?
This is just a sketch of the calculation of the conjugacy classes. As I said in my comment, $t$ conjugates $s$ to $s^3$, $s^2$ to $s^6$, and it centralizes $s^4$, so the conjugacy classes contained in $\langle s \rangle$ are $\{ 1 \}$, $\{ s^4 \}$, $\{s,s^3\}$, and $\{s^2,s^6\}$.
Also, $sts^{-1}= ts^3s^{-1} = ts^2$, $s(ts^2)s^{-1} = ts^4$ and $s(ts^4)s^{-1}= ts^6$, so another class is $\{ t,ts^2,ts^4,ts^6 \}$, and similarly there is a class $\{ ts,ts^3,ts^5,ts^7 \}$, making $7$ classes altogether.