Solution
= Solution
There are twelve edges and the action on them is transitive, so an edge stabilizer has order
$$
\frac{48}{12}=4.
$$
For an edge parallel to the $z$-axis with midpoint $(1,1,0)$, its stabilizer is generated by interchanging $x$ and $y$ and by replacing $z$ by $-z$. These are commuting involutions, so
$$
\boxed{G_e\cong C_2\times C_2}.
$$