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.

  • $\begingroup$ 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? $\endgroup$ – hhattiecc Nov 18 '16 at 11:18

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