The dodecahedron has 20 vertices, 30 edges and 12 faces, each of which is a regular pentagon. To construct a dodecahedron whose sides have length 1, we start with a cube of side t=(1+Ö5)/2 and attach to each face a "tent" with edges of length 1, as shown in the diagram below.
It is not obvious that the trapezia line up properly with the triangles to give flat pentagons, but a proof is given in PMA334. We can actually inscribe 5 different cubes in a dodecahedron; they are shown in 5 different colours below. You should look carefully at this diagram and convince yourself that the green lines (for example) really do form a cube, and that each face contains exactly one line of each colour.
The symmetry group acts on the set of these cubes, giving a homomorphism f:Symm(Dodec)->S5. By joining each vertex to the opposite one, we obtain 10 different lines of 3-fold rotational symmetry. We can twist around each of these axes by an angle of 2p/3 or 4p/3, giving 20 different rotations of order 3, each of which gives a 3-cycle. By joining the centre of each face to the centre of the opposite face, we obtain 6 different lines of 5-fold rotational symmetry, giving 24 rotations of order 5, each of which gives a 5-cycle. By joining the centre of each edge to the centre of the opposite edge, we obtain 15 different lines of 2-fold rotational symmetry, each of which gives a transposition pair. It can be shown using this that f is actually an isomorphism Symm(Dodec)->A5. The three types of rotation axis are shown below.
The dual of a dodecahedron is an icosahedron.