I'm trying to understand values of the character of the representation of $S_4$ corresponding to the symmetries of a regular tetrahedron (whew, that's a mouthful!). One illustrative video is found here, where the different conjugacy classes of $S_4$ acting on the regular tetrahedron are discussed.
What I'm struggling with is understanding how you can deduce how the $x, y, z$ values are mapped to each other in the case of a single two cycle $(1, 2)$, two two cycles $(1, 2)(3, 4)$ and the four cycle $(1, 2, 3, 4)$. Let's consider for example the single two cycle $(1, 2)$: We'd like to flip (I'm referring to the illustrations of the video) the swap the vertices of the lower right edge of the tetrahedron while keeping $3$ and $4$ fixed. Does the lecturer deduce that $x\mapsto x$ by turning the tetrahedron over, so that the vertex $4$ points downwards? And then there are the other cycle types I mentioned...