A dihedral group Dn is the group of symmetries of a regular polygon with n sides, including both rotations and reflections.
A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries.
Dn has 2n elements.
Dihedral groups Dn are non-Abelian permutation groups if n > 2.
A Dihedral group Dn is a subgroup of Sn, which is a symmetric group with n vertices.
Matrix representation
If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication.
Dihedral Group Graph Automorphism Demonstrations of Graph Automorphism