A permutation is an arrangment of object. A permutation group is all the possible arrangments of a set of objects. The only permutation groups that can be modeled in the plane are those of orders one, two, and three. If you don't believe this, try getting all of the permutations of four objects using a square, a rotation of order four and a reflection. It can't be done.
Permutation groups contain only reflections and rotations.
The permutation group of order one is tivial. The permutation group of order two consists of 12 and 21. If we have three objects, then the permutation group has order six. Mathematicians usually designate this group S3. This groupg even has a subgroup that is normal.