Symmetries of a dodecahedron Suppose we want to measure order of the group symmetries of a dodecahedron, and we know that If $G$ is a group and $S$ is a set on which $G$ acts and $s\in S$, then Order of G=(Order of stabiliser of s) x (Order of Orbit of s). Using this find order of Group of symmetries of a dodecahedron.

Hint: let $s$ be one of the faces of the dodecahedron. The _orbit_ of $s$ is the set of faces to which $s$ can be mapped under the action of the group; can you figure out how many elements $s$ can be mapped to? (Don't forget to count $s$ itself) Likewise, the _stabilizer_ is the set of symmetries that leave $s$ 'in place'; can you figure out how many of those there are?

As another check on this, you could choose to let $s$ be one of the _vertices_ of the dodecahedron rather than its faces; you should find the same result by multiplying the size of the orbit by the size of the stabilizer.