In mathematics, a dihedral group is the group of symmetries of a regular polygon with sides, including both rotations and reflections. This Demonstration shows the subgroups of , the dihedral group of a square.

A group is a set together with a binary operation on , i.e., a function to (called the group law of ) that combines any two elements and to form another element, denoted or . To qualify as a group, the set and operation, (, ), must satisfy four requirements known as the group axioms: closure, associativity, identity element, and inverse element. If , then the group is commutative or Abelian.

In this Demonstration, the group law is the composition of permutations of the set . For example, .