It is known that the set of orientation preserving symmetries of a cube is isomorphic to the symmetric group of four letters, $S_4$. Since each of the different orientations can be described by a permutation matrix with signed non-zero elements such that the determinant is equal to one (to preserve orientation), I was wondering how you could construct the isomorphism explicitly between the set of these matrices and the set $S_4$. To be explicit, the rotation matrix $\begin{bmatrix}1 & 0 & 0\\0 & 0 & 1 \\0 & - 1 & 0\end{bmatrix}$ corresponds to the permutation $(1)(23)$ where the sign of $2$ is flipped and in principle we can denote this signed permutation by $((1)(23), (+, -, +))$. Therefore I was wondering how you could construct and an isomorphism between these signed permutations and $S_4$.
While I've seen a clever way to label the vertices of the cube as $1,2,3,4$ in order to represent any combination of rotations as a permutation of the four elements, I'm still curious on how you could establish explicitly the that $S_4$ is isomorphic to particular signed permutations of $S_3$ (which in turn correspond to the said signed permutation matrices).
I don’t know what you mean by “signed permutations of $S_3$ (which correspond to the said signed permutation matrices)”; I’ll assume that that’s just a wrong formulation and what you actually mean is what you wrote further up.
To map the orientation-preserving symmetries of the cube to signed permutation matrices, place the cube with its centre at the origin of a Cartesian coordinate system and express the symmetries as matrices in that coordinate system.
To map the symmetries to elements of $S_4$, consider how they permute the four body diagonals (ignoring inversions).
For instance, the rotations through $\frac\pi2$ about the three coordinate axes map to the matrices $\pmatrix{1&0&0\\0&0&1\\0&-1&0}$, $\pmatrix{0&0&-1\\0&1&0\\1&0&0}$ and $\pmatrix{0&1&0\\-1&0&0\\0&0&1}$, and for a certain labelling of the body diagonals they map to the permutations $(1234)$, $(1243)$ and $(1423)$ in $S_4$, respectively. You can generate all the other correspondences from those, or read them off the body diagonals, as you prefer.