Dihedral group of order 8

In mathematics, D4 (sometimes alternatively denoted by D8) is the dihedral group of degree 4 and order 8.

[1][2] As an example, we consider a glass square of a certain thickness with a letter "F" written on it to make the different positions distinguishable.

In order to describe its symmetry, we form the set of all those rigid movements of the square that do not make a visible difference (except the "F").

Again, after performing this movement, the glass square looks the same, so this is also an element of our set and we call it b.

[3] This version of the Cayley table shows that this group has one normal subgroup shown with a red background.

Cycle graph of Dih 4
a is the clockwise rotation
and b the horizontal reflection.
Cayley graph of Dih 4
A different Cayley graph of Dih 4 , generated by the horizontal reflection b and a diagonal reflection c
The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13) . The numbers in this table come from numbering the 4! = 24 permutations of S 4 , which Dih 4 is a subgroup of, from 0 (shown as a black circle) to 23.