A tetrahedron has $12$ rotational symmetries and $24$ isometries in total. This means that the group of isometries is isomorphic to $S_4.$
If we denote the vertices of a tetrahedron as $1$, $2$, $3$, and $4,$ then what will be the interpretation of the transformation that permutes the vertices as follows: $$ 1 \to 2 \to 3 \to 4 \to 1, $$ as a cycle of length $4?$
I mean, in terms of rotations and reflections.
A $4-$cycle corresponds to the compose of a rotation and a symmetry. Indeed, you can write $$(1 \text{ } 2 \text{ } 3 \text{ } 4)= (1 \text{ } 2 \text{ } 3)(3 \text{ } 4) $$
and it is easy to see that a $3-$cycle is a rotation and a transposition is a planar symmetry.