This is an example of bielliptic surface on page 84 from Beauville's book "Complex algebraic surfaces".
Let $\rho^3=1$, $\rho\neq1$ and $F_\rho=\mathbb{C}/(\mathbb{Z}+\rho\mathbb{Z})$, $G=\mathbb{Z}/6\mathbb{Z}$ be a group of translations of an elliptic curve $E$ acting on $F_\rho$ by $x\mapsto -\rho x$. Let $S=(E\times F_\rho)/G$.
1) How to compute the canonical class $K_S$ of $S$?
2) How to show that $2K_S\neq0$, $3K_S\neq0$ but $6K_S=0$?
There are some explanations of this in the Beauville's book but I don't understand the details, so I'd very much appreciate someone's help.
$\textbf{Update}$
Let me explain my difficulties.
Beauville proves that $\text{dim}\,H^0(S, mK_S)=\text{dim}\,H^0(F, \omega_F^{\otimes m})^G=\text{dim}\,H^0(F/G, \mathcal{L}_m)$, where $\mathcal{L}_m=\omega_{F/G}^{\otimes m}(\sum\limits_{P\in F/G}[m(1-\frac{1}{e_P})]P)$, $[x]$ is an integral part of $x$.
If $F/G\cong\mathbb{P}^1$ then $\mathcal{L}_m$ is determined by its degree $$\text{deg}\,\mathcal{L}_m=-2m+\sum\limits_{P\in F/G}[m(1-\frac{1}{e_P})].$$ By Riemann-Hurwitz formula this degree should be zero for $m=6$.
I don't know how to prove that $F/G\cong\mathbb{P}^1$ and how to find ramification points and ramification indices.