Is there a technique to derive the groups of rotations of various objects?

Question

Apart from simply memorising them or being able to visualise them on the spot and jot them down is their any way to derive the group of rotations for various shapes in 3-d. (2-d is easy enough to just draw them and derive so never mind those)

Answer 1

Rotations of a symmetric 3D shape can generally be broken down into: 1) Considering the number of faces 2) The number of vertices per face.

For example, with the icosahedron, the shape has 20 faces, all with three vertices. The rotation group clearly has size 3x20=60. The elements of the group consist of permuting the vertices in ways as to not tear or reflect the shape

Answer 2

I saw this thread earlier when it was overrun by trolls (a worrying trend that I've noticed on this website as of late). A good general method is basically to look at the vertices and faces of the shape, and permute the faces, and see what information this gives you.