proper rotation axis (Cn)

Picture a three-bladed fan or a child's pinwheel. Give it a third of a turn — 120 degrees — and it looks exactly the way it did before. You could keep doing that and never tell. The line you spin it around is a proper rotation axis, written Cn, and it is the most familiar kind of molecular symmetry: an axis you can rotate the molecule about by a definite angle and get an identical picture.

The subscript n tells you the order: rotation by 360/n degrees brings the molecule back onto itself. A C2 axis means a 180-degree turn works; a C3 means 120 degrees works; a C4 means 90 degrees. One Cn axis generates several operations — for a C3 you get a 120-degree turn (C3), a 240-degree turn (C3 squared), and a full 360-degree turn (which equals the identity E). A molecule may possess many axes at once. The axis with the largest n is called the principal axis and, by convention, is taken to point along the vertical (z) direction; it anchors the rest of the symmetry description.

Proper rotations are called "proper" because they are physically realizable rigid motions — you really can turn the molecule that way without taking it apart. This sets them apart from improper rotations (Sn) and reflections, which involve a mirror image and so cannot be done by simply moving the object in space. Counting the highest-order proper axis is the very first step in assigning a molecule to its point group, which is why the Cn axis sits at the start of every classification flowchart.