Symmetries of a Square and an Equilateral Triangle

The notation is based on that used in [1]. Rotations are counterclockwise and composition is right-to-left, that is, in , acts first. The reflection over the horizontal axis through the center of a figure is denoted ; stands for a reflection over the central vertical axis. The reflection over the "main" diagonal (upper-left to lower-right) of a square is denoted ; is the reflection over the other diagonal. Similarly, when the figure is set to "triangle," denotes the axis of reflection that cuts through the upper-left edge and lower-right vertex of the triangle, and is the axis of reflection sloping the other way. The elements of the group in the Cayley table are listed with rotations preceding reflections.

Reference

[1] J. Gallian, Contemporary Abstract Algebra, 7th ed., Belmont, CA: Brooks/Cole, Cengage Learning, 2010.