State and prove the orbit-stabilizer theorem.

Let $G$ be the group of all symmetries of a regular octahedron, including both orientation-preserving and orientation-reversing symmetries. How many symmetries are there in the group $G$ ? Let $D$ be the set of straight lines that join a vertex of the octahedron to the opposite vertex. How many lines are there in the set $D$ ? Identify the stabilizer in $G$ of one of the lines in $D$.