12. Let $\alpha = \begin{pmatrix} 1 & 2 & 3 & 4 \ 2 & 3 & 4 & 1 \end{pmatrix}$ and $\beta = \begin{pmatrix} 1 & 2 & 3 & 4 \ 2 & 1 & 4 & 3 \end{pmatrix}$, and let $G = \{e, \alpha, \alpha^2, \alpha^3, \beta, \alpha\beta, \alpha^2\beta, \alpha^3\beta\}$ be the subgroup of $S_4$ generated by $\alpha$ and $\beta$. $G$ acts on the set $X = \{1, 2, 3, 4\} \times \{1, 2, 3, 4\}$: if $\sigma \in G$, then $\sigma(i, j) = (\sigma(i), \sigma(j))$.
a. For $x = (1, 1)$, $y = (1, 3)$, and $z = (1, 4)$ in $X$, find the orbits $O_x$, $O_y$, $O_z$, and the stabilizers $G_x$, $G_y$, $G_z$.
b. Find the partition of $X$ given by the orbits of $G$.
c. For elements $g = \alpha^2\beta$ and $h = \alpha^3\beta$ of $G$, find the fixed sets $X_g$ and $X_h$.
