I am confused about boundary maps of a CW complexes. For example, let $X$ be $\mathbb{C}\times \mathbb{R}$. $t_1,t_2,\alpha$ are the transformations on $X$ such that
$$t_1:(z,x)\mapsto (z+1,x)$$
$$t_2:(z,x)\mapsto (z+\xi,x)$$
$$\alpha:(z,x)\mapsto (\omega z,x+1)$$
where $\xi=\frac{1+\sqrt{-3}}{2},\omega=\frac{-1+\sqrt{-3}}{2}$. Let $G$ be the group generated by $t_1,t_2,\alpha$. I want to find the homology groups of $X/G$. I tried to use the CW complexes. I arranged 2-cells as following. Note that the rest 3 faces are identified with $e^2_i(i=1,2,3)$.
My question: What is $d(e^3)$? The right and left sides, front and back sides have the "different" orientation, so $d(e^3)$ doesn't have the terms $e^2_2,e^2_3$. But the above and below faces' orientation is twisted. So I confused weather $d(e^3)$ is $2e^2_1$ or $0$. Please give me some advice.