ABSTRACT ALGEBRA- Permutation proof?

Consider a regular n-gon for n >=3. Each way that two copies of such an n-gon can be placed, with one covering the other, corresponds to a certain permutation of the verticies.The set of these permutations in a group, the nth dihedral group (D subscript n), under permutation multiplication. Find the oder of this group D sub n. Argue geometrically that this group has a subgroup having just half as many elements as the whole group has.