Example: Tetrahedral Group

Consider an equilateral tetrahedron as shown. The Tetrahedral Group consists of all ways of rotating and reflecting the tetrahedron so that its orientation is preserved. There are a total of twelve ways of doing this, so that the Tetrahedral Group has twelve elements. Each of these twelve elements can be obtained by a combination of two generating elements, namely a "swap" S and a "twist" T.

A swap can be thought of as concentrating on an edge of the tetrahedron and turning the tetrahedron around by 180° so that the edge stays in the same place but the vertex labels at each end of the edge have been swapped. In the process of doing this, the other two vertices are swapped as well. A twist can be thought of as holding one vertex of the tetrahedron and twisting the tetrahedron so that the other three vertices cycle.

Consider that the tetrahedron is now embedded in R³ as shown on the left. Note that the placement is such that the points on edge AD have the same value of y, points on edge AC have the same value of x, and points on edge AB have the same value of z.

The "swap", S, is performed by concentrating on the edge AB and turning the tetrahedron by 180° so that the edge remains in the same place but the labels at both ends of the edge have been swapped. In the process of doing this, we also swap the other two vertices, B and C. We can visualise this operation by "swinging" the cube enclosing the tetrahedron around the z-axis by 180°, so that the cube looks exactly the same as before (as does the whole tetrahedron, so orientation is preserved), but vertices A and B have been swapped. In the process of doing this, we will also have swapped vertices C and D. In matrix form, the process of "swinging" around the z-axis by 180° can also be described as reflecting the cube in the x and y axes. Therefore, we can represent the swap S by the following matrix:

Consider that the "twist", T, is implemented at any stage by moving the tetrahedron so that the edge which is in the x-z plane moves to the edge which is in the x-y plane; the edge which is in the y-z plane moves to the edge which is in the x-z plane; and the edge which is in the x-y plane moves to the edge which is in the y-z plane. For the representation in the above diagram, performing T would involve moving AD to CD, moving CD to AC, and moving AC to AD. This corresponds to the permutation (ADC). The important point is no matter what the labels are, we do exactly the same "movement" when performing T. In matrix representation,

Try entering the above two generators into the Applet and see what happens. You should obtain the group shown below. After you obtain the group, find the subgroups and the Hasse Diagram. Try values of 80 for S and -40 for G to obtain a "better" diagram.

Elements of the Tetrahedral Group in matrix representation and permutation representation
(S = (AB)(CD) and T = (ADC) in permutation representation)