Example

Construct a "multiplication table" for the symmetries of a rectangle.

Your answer may differ from what follows depending on: how you labeled your transformations, in what order you placed your transformations, which transformation you applied first, and so on... Amazingly however, the overall structure of your answer won't change even if you made different choices than were made below.

Consider the possible ways to reposition a rectangle, subject to the restraint that we must end up in the same area our rectangle once occupied:

 Identity, I Rotate 180°, R Vert. Axis Flip, V Horiz. Axis Flip, H

Notice, we fail to have a rotation of 90 degrees, as such a rotation would make the rectangle fall outside of its original area occupied.

With a little help from an index card whose corners are labeled: A, B, C, and D; we can find the net effect of applying any two of these transformations together. (Assume the column below indicates the first transformation applied.)

I R V H
I I R V H
R R I H V
V V H I R
H H V R I