Dihedral Group

The dihedral group $D_{2n}$ consists of the rotational and reflectional symmetries of a regular $n$-gon. In terms of permutations of a ciphertext alphabet of length $n$, this can be thought of as the group of rotations of the alphabet and of its reverse (or decimation by $n - 1$). One physical implementation of group autokey ciphers with this as the underlying group are mechanisms based on a cipher disk where each plaintext letter can rotate the disk by some particular amount (fixed for each plaintext alphabet letter) and can also optionally flip the disk over. The same symbols are written on both sides of the disk, but when the disk is flipped over, the following rotations appear to have the opposite effects. This lack of commutativity causes chaining conflicts. However, it has been shown that the eyes cannot be a dihedral cipher (with group $D_166$).