The tetrahedron inside a cube.

 

The cartesian axes of the cube (x, y, z) are the C2 axes of the tetrahedron. They also serve as S4 axes. The body diagonals of the cube are the C3 axes. There are four such axes. The plane bisecting a pair of cartesian axes is a mirror plane. There are six such planes.

 

You should see what happens to the x, y, z axes and the vertices (labelled 1,2,3,4) upon each symmetry operation.

 

For example, C3(2) anticockwise will have the following effect:

 

x --> y

y --> z

z --> x

1 --> 3

3 --> 4

4 --> 1 (2 does not move, since it is on the axis)

 

Remembering that C3 and S4 elements, each generate two separate symmetry operations, the full list of operations of the tetrahedron arranged in classes will be as follows: E, 8C3, 3C2, 6S4, 6sd.