A3 (symmetric group S4)  is a group that consists of 24 elements. The action of these elements can be represented as reflexions and rotations in the three dimensional space. Therefore, this group can transform one single point of space into 24 different points. The result is a geometrical object that consists of 24 vertices.

However, if the point that you start with exactly lies on such  plane of reflexion, it won't be transformed into another point. It turns out that in such a case, exactly half of the 12 elements remain passive and one ends up with a geometrical object that consists of 12 vertices. Similarly, geometrical objects with 6 and 4  vertices can be obtained. Starting from the most general representation with 24 vertices, all other representations are obtained by identifying vertices. 

Apart from the tetrahedron, all polyhedra are also representations of B3. The dual of the tetrahedron is a tetrahedron again. And the merger of the tetrahedron with its dual yields a cube.