Polygon Dihedral Groups of Order 2n Consider a regular polygon P_{n} with n sides. The set of rigid motions from P_{n} to itself forms a group under composition of functions. The group consists of n rotations and n reflections. The rotations are about the center of P_{n} through angles that are multiples of 360/n. The axes of the reflections connect each vertex of P_{n} to the midpoint of its opposite side when n is odd. When n is even, half the axes connect pairs of opposite vertices and half connect the midpoints of opposite sides. The group, denoted D_{n}, is called the dihedral group of order 2n. 

The applet displays a polygon with an adjustable number of sides. Rotations and reflections may be applied using the buttons. The axis of reflection may also be adjusted. Click on an element in the 'Current' column to set the current element. Clicking in the 'Compose' column applies the element clicked on to the current element. 