Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 8D / b / ii / Solution

There are twelve edges and the action on them is transitive, so an edge stabilizer has orderFor 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