Could someone please explain the difference between the group of all icosahedral symmetries and S5? I know that the former is a direct product, but don't they work the same? Say I have an icosahedron, why wouldn't S5 work as a description of its symmetries? Thank you very much.
Added: When counting the symmetries of a platonic solid, in this case the icosahedron, Does it include reflecting along a plane cutting through the solid, in a sort of turing itself inside out reflection? I read that the symmetries counted should be "orientation-preserving". What does that mean?