Stabilizer of diagonal of a cube

Question

The group of rotational symmetries of a cube, $G \cong S_4,$ acts transitively on the set of four diagonals of a cube. Let's denote by $X=\{1,2,3,4\}$ the set of four diagonals. Clearly $|\text{Orb }(1)| =4,$ and since $|G|=24,$ by orbit-stabilizer theorem, $|\text{Stab } (1)|=6.$

However, there are only three possible rotations (angle $0, 2\pi/3,4\pi/3$) through a diagonal. What are the other rotations?

I know that there are three more rotations corresponding to $(23),(34),(24)\in S_4,$ but it is very hard to picture them. What is the axis of rotation?

Answer 1

There are three more rotations that swap the ends of the fixed diagonal. Take the plane containing any two diagonals. Rotate it through $\pi$, and the other two diagonals are swapped.

Answer 2

You get the other three rotations by rotation by $\pi$ about the lines through opposite edges which does not share a common vertex with the diagonal.

Concretely, if the cube is $\{\pm 1\}^3$ and our diagonal is $(-1,-1,-1)\text{-}(+1,+1,+1)$, we get three rotations by $\pi$ about lines $(0,+1,-1)\text{-}(0,-1,+1)$ and similar.