I can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a regular triangle. But I need to know the elements of the group and I don't have not only a physical regular tetrahedron but also my imagination is not strong. Videos like $1$ or $2$ didn't help, because I can't understand for example what happens to the other $2$ points when the top point and one of the surface points is exchanged.
I would appreciate any simple clear explanation.
Added – For next permutations (with order $3$) now simply we consider the ‘surface’ under the other three-points, i.e. each time $4$ on top is replaced by $1$ or $2$ or $3$. And, we perform rotations as we were doing when $4$ was on top. So we will get:
‘$4$’ on top : ${\{(123),(132)}\}$
‘$1$’ on top : ${\{(234),(243)}\}$
‘$2$’ on top : ${\{(134),(143)}\}$
‘$3$’ on top : ${\{(124),(142)}\}$.
But what about $(12)(34)$, $(13)(24)$ and $(14)(23)$? How to get them?
Best Answer
One of them
Use different axes (connecting the midpoints of two opposite edges of the tetrahedron) to get the other products of two disjoint 2-cycles as 180 degree rotations.
Here is the same animation with the cube surrounding the tetrahedron shown as a wireframe. The axis of rotation joins the centers of two opposite faces of the cube