# Rotational Symmetry Group for Edges, Vertices, Faces

I'm having difficulty finding what the symmetry groups of rotation are for certain 3D shapes.

I'm interested in finding the different colourings (when you have $k$ colours, say) of a shape's edges, vertices and faces up to rotation.

I've been told the following: I know that for a regular tetrahedron, the symmetry group up to rotation for it's faces is isomorphic to $A_4$, the even permutations in $S_4$. For a cube's vertices it's isomomorphic to $S_4$.

My question is: is there a general 'rule' for knowing what the symmetry group of rotation is isomorphic to? I've looked at many webpages explaining what each individual rotation is, but I find it quite difficult to immediately visualise this in my head (and especially when rotating 3D shapes to preserve colourings isn't something I find very easy to draw/keep track of).

My original intuition was telling me this:

1) Identify the number of 'things' (vertices, faces, edges).

2) Are these 'things' fixed together, e.g. if they're edges, every edge is connected in a rigid manner. If we're going to preserve colourings then we must also preserve this structure which determines how they lie respective to one another.

3) If the answer to 1 is $n$ and the answer to 2 is yes, then most likely $A_n$, if the answer to 1 is $n$ and the answer to 2 is no, then most likely $S_n$.

But then I realised that for a cube's edges this way too large.

How would you go about understanding what the rotational symmetry group for a tetrahedron's edges is, for example.

• Also, if we know one element in this group has cycle shape (3,3) for example, can we say that all cycle shapes of (3,3) are in the rotational symmetry group? – hhattiecc Nov 18 '16 at 11:18