What is the difference between these groups?

Question

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?

Answer 1

Hint: If $g$ is the symmetry that send each point to it's opposite point in the icosahedron, then show $\{1,g\}$ is a normal subgroup of the group of symmetries. Show that $S_5$ does not have any normal subgroup of order $2$.

Answer 2

Two groups are treated as same if there is an isomorphism between them.A simple reason why $S_5$ cannot be used to describe the symmetries of an icosahedron(whose group of symmetries we will call $I_h$) is that its structure is fundamentally different from that of $I_h$. For starters, $S_5$ cannot be expressed as a direct product of two groups unlike $I_h$, for which it is possible to do so. So $S_5$ cannot be isomorphic to $I_h$.