Tetrahedra consists of four vertices which are all connected to each other by edges. These edges form the sides of four equilateral triangles. Since each vertex is connected to every other vertex by an edge the vertices can be arranged in any way and still have all the edges connecting the same vertices.
If a number is assigned to each vertex of a regular tetrahedron, any symmetry that maps the tetrahedron onto itself is easiliy represented as a permutation of the vertices. A single transposition will be the equivalent to a reflection, as the orientation is reversed. These reflection symmetries are discussed in the next part. Below I show how any pair of transpositions is equivalent to a rotation about the center of the tetrahedron. If the transpositions are disjoint, the rotation is about the line connecting the centers of the edges of the trasposed pairs. If one of the vertices is fixed in both transpositions, the rotation is about the axis through that vertex.
Two disjoint transpositions represent a rotation of π about the axis through the midpoints of the edges that connect the transposed pairs. Since these are disjoint, the order does not matter.
(12)(34)=(34)(12)
Two nondisjoint permutations represent a rotation of 2π⁄3 about the axis through the fixed vertex.
(23)(34)=(234)
Reversing the order of the transpositions represents a rotation in the other direction, or a combination of two rotations in the original direction. Thus it is also the inverse of a rotation in the opposite direction.
(34)(23)=(243)
(243)=(234)(234)
(234)(243)=(243)(234)=1
Note that the vertices used in the above rotations can be assigned in any manner and the symmetry will still be valid.
At left are the seven elementary rotations. For ease of notation, I will define R12 to represent the rotation of π about the axis through the center of edge 12. This rotation then is equivalent to the permutation (12)(34). R13 and R14 are similarly defined.
R1 will represent a counter-clockwise rotation of 2π⁄3 about the axis passing through vertex 1, facing the center through 1. This is equivalent to the permutation (234). R2, R3, and R4 will similarly represent the rotations about vertices 2, 3,and 4.
R12, R13, and R14 are their own inverses, so R122 = E. On the other hand, R1, ..., R4 all have order 3, so R12, ..., are also possible rotations.
These 11 rotations, along with the identity, represent all possible rotations of regular tetrahedra. The group tables below show the possible permutation cycles of the vertices and the corresponding rotations.
| E | R12 | R13 | R14 | R12 | R22 | R32 | R42 | R1 | R2 | R3 | R4 | |
| E | E | (12)(34) | (13)(24) | (14)(23) | (243) | (134) | (142) | (123) | (234) | (143) | (124) | (132) |
| R12 | (12)(34) | E | (14)(23) | (13)(24) | (142) | (123) | (243) | (134) | (132) | (124) | (143) | (234) |
| R13 | (13)(24) | (14)(23) | E | (12)(34) | (123) | (142) | (134) | (243) | (143) | (234) | (132) | (124) |
| R14 | (14)(23) | (13)(24) | (12)(34) | E | (134) | (243) | (123) | (142) | (124) | (132) | (234) | (143) |
| R1 | (234) | (124) | (132) | (143) | E | (13)(24) | (14)(23) | (12)(34) | (243) | (142) | (123) | (134) |
| R2 | (143) | (132) | (124) | (234) | (13)(24) | E | (12)(34) | (14)(23) | (123) | (134) | (243) | (142) |
| R3 | (124) | (234) | (143) | (132) | (14)(23) | (12)(34) | E | (13)(24) | (134) | (123) | (142) | (243) |
| R4 | (132) | (143) | (234) | (124) | (12)(34) | (14)(23) | (13)(24) | E | (142) | (243) | (134) | (123) |
| R12 | (243) | (123) | (134) | (142) | (234) | (132) | (143) | (124) | E | (14)(23) | (12)(34) | (13)(24) |
| R22 | (134) | (142) | (243) | (123) | (124) | (143) | (132) | (234) | (14)(23) | E | (13)(24) | (12)(34) |
| R32 | (142) | (134) | (123) | (243) | (132) | (234) | (124) | (143) | (12)(34) | (13)(24) | E | (14)(23) |
| R42 | (123) | (243) | (142) | (134) | (143) | (124) | (234) | (132) | (13)(24) | (12)(34) | (14)(23) | E |
| E | R12 | R13 | R14 | R12 | R22 | R32 | R42 | R1 | R2 | R3 | R4 | |
| E | E | R12 | R13 | R14 | R12 | R22 | R32 | R42 | R1 | R2 | R3 | R4 |
| R12 | R12 | E | R14 | R13 | R32 | R42 | R12 | R22 | R4 | R3 | R2 | R1 |
| R13 | R13 | R14 | E | R12 | R42 | R32 | R22 | R12 | R2 | R1 | R4 | R3 |
| R14 | R14 | R13 | R12 | E | R22 | R12 | R42 | R32 | R3 | R4 | R1 | R2 |
| R1 | R1 | R3 | R4 | R2 | E | R13 | R14 | R12 | R12 | R32 | R42 | R22 |
| R2 | R2 | R4 | R3 | R1 | R13 | E | R12 | R14 | R42 | R22 | R12 | R32 |
| R3 | R3 | R1 | R2 | R4 | R14 | R12 | E | R13 | R22 | R42 | R32 | R12 |
| R4 | R4 | R2 | R1 | R3 | R12 | R14 | R13 | E | R32 | R12 | R22 | R42 |
| R12 | R12 | R42 | R22 | R32 | R1 | R4 | R2 | R3 | E | R14 | R12 | R13 |
| R22 | R22 | R32 | R12 | R42 | R3 | R2 | R4 | R1 | R14 | E | R13 | R12 |
| R32 | R32 | R22 | R42 | R12 | R4 | R1 | R3 | R2 | R12 | R13 | E | R14 |
| R42 | R42 | R12 | R32 | R22 | R2 | R3 | R1 | R4 | R13 | R12 | R14 | E |
Click here for a larger version of the tables.
These twelve permutations are clearly even permutations of four elements, and thus the set of rotations is isomorphic to A(4).