We know that the total symmetry group of tetrahedron is $S_4$. I tried to realize these 24 symmetries, but failed to ''realize'' six of them, which correspond to the $4$-cycles $(a \,b\,c\,d)$ (being permutations of vertices).
Question: What is an easiest way to "realize" these six symmetries in 3-space, which common undergraduates can understand.
The other six are neither reflections nor rotations. They reverse handedness, so they need a reflection.
Take two edges - 13 and 24. Take their midpoints 1A3 and 2B4. Draw an axis between A and B. Rotate 90 degrees about AB, then reflect in the plane that perpendicularly bisects AB.