The tetrahedron has 4 vertices, 6 edges and 4 faces, each of which is an equilateral triangle. There are 6 planes of reflectional symmetry, one of which is shown on the below. Each such plane contains one edge and bisects the opposite edge (this gives one plane for each edge, hence 6 planes). Reflection in a plane fixes two of the vertices and exchanges the other two, so the corresponding vertex permutation is a transposition.
There are 4 lines of 3-fold rotational symmetry, each of which passes through a vertex and the centre of the opposite face (giving one line for each vertex). These are shown below in red. The corresponding vertex permutations are 3-cycles. There are also 3 lines of 2-fold rotational symmetry, shown in green. Each one joins the centres of an opposite pair of edges. The corresponding edge permutations are transposition pairs, in other words they have the form (a b)(c d) where a,b,c and d are all different.
If we start with a tetrahedron (such as the wire frame below) and find the centres of all the faces we get the vertices of a new tetrahedron (the coloured one). Thus, the tetrahedron is self-dual.