What are all the elements of the group of symmetries of a regular tetrahedron?

Solution 1:

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

Solution 2:

Consider the set of half edges of the tetrahedron. It is easy to convince yourself that the symmetry group of the solid allows you to move from one such half edge to any other in exactly one way. Since there are six edges, there are twelve half edges, and this implies that the group of symmetries has exactly 12 elements.

Now the group obviously permutes the four vertices. Since there is exactly one subgroup of order 12 in $S_4$, namely the alternating group, it must be our group.