Group of orientable symmetries of the 3-dimensional unit cube. Here is the question I am trying to solve:
Let $G$ be the group of orientable symmetries of the $3$-dimensional unit cube. Prove that $G$ is isomorphic to the symmetric group $S_4.$
I also got the following hint:
The unit cube has four diagonals that pass through the opposite corners, and $G$ acts on the set of diagonals.
Still, I do not know how to solve this problem. I have the following questions:
1- What is the meaning of orientable symmetry?
2- What is the relation between the action of $G$ and the isomorphism I should find?
Could someone help me answer these questions and solve the problem, please?
EDIT:
Do we actually need to know all the rotations of the cube and conclude from them the corresponding permutations of the vertices to write the isomorphism in this question?

Your book's hint is to consider the four diagonals of the cube (the long diagonals, lines passing through a pair of antipodal vertices), which are permuted by every symmetry of the cube, and particularly by every rotation. In the hope a sketch helps:

Enumerating all the rotations is not necessary, but it might be a good exercise depending on your background and interests. Rotations of the cube can be described, for example, by

*

*Geometric action as rotation about some axis;

*Effect as a permutation on the set of four diagonals;

*As signed $3 \times 3$ permutation matrices of positive determinant: $3 \times 3$ real matrices whose entries are all $0$, $1$, or $-1$, that have precisely one non-zero entry in each row and each column, and whose determinant is $1$.

With an eye toward minimizing computation, we can show successively that

*

*Every rotation of the cube permutes the diagonals (diagonals map to diagonals, and distinct diagonals map to distinct diagonals).

*The mapping from rotations to permutations is a homomorphism (performing successive rotations effects performing successive permutations on the diagonals).

*The mapping from rotations to permutations is injective (no non-identity rotation fixes every diagonal).

*The number of cube rotations is equal to the number of permutations of the set of diagonals.

To count the number of rotations without enumerating possibilities, it helps to think about:

*

*If we fix a vertex $0$, how many locations are there for the image of $0$ under a rotation of the cube?

*How many rotations fix the vertex $0$ (or any particular vertex)?

