Glue two dodecahedra together along a pentagonal face and find the rotational symmetry group of this solid. What is its full symmetry group?
The Attempt at a Solution
Since this clearly is a finite subgroup of SO3, it must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron, cube, or icosahedron.
I have only found two types of axes of rotational symmetry so far: one that runs through the centers of the faces at the ends of the figure and these have order five.
Or we can run an axis along the face where the two dodecahedra are joined and rotate about that by pi. There are 5 of this type of axis. I can't seem to find any other ones, but with just these, the rotational symmetry group is not isomorphic to any of the ones mentioned above.
And I'm not entirely sure how a full symmetry group differs from a rotational one.