Element structure of dihedral group:D8

We denote the identity element by $$e$$. The dihedral group $$D_8$$, sometimes called $$D_4$$, also called the of order eight or the dihedral group acting on four elements, is defined by the following presentation:

$$\langle x,a| a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle$$



The row element is multiplied on the left and the column element is multiplied on the right.



Elements
Below, we list all the elements, also giving the interpretation of each element under the geometric description of the dihedral group as the symmetries of a 4-gon, and for the corresponding permutation representation (see D8 in S4). Note that for different conventions, one can obtain somewhat different correspondences, so this may not match up with other correspondences elsewhere. Note that the descriptions below assume the left action convention for functions and the corresponding convention for composition, and hence some of the entries may become different if you adopt the right action convention.:

Commutator map
Because of the fact that the inner automorphism group is an elementary abelian 2-group, it does not matter which of the two definitions of commutator map we choose ($$[\alpha,\beta] = \alpha^{-1}\beta^{-1}\alpha\beta$$ or $$[\alpha,\beta] = \alpha\beta\alpha^{-1}\beta^{-1}$$) -- they are both the same map.

In fact, the commutator map sends a pair of elements to $$e$$ if they commute and to $$a^2$$ if they don't commute.

General description


The equivalence classes up to automorphisms are:



Interpretation as dihedral group
Below, we consider the conjugacy class structure in terms of the interpretation ofthe group as a dihedral group of degree $$2n$$, where $$n = 4$$ is even:

Interpretation as unitriangular matrix group
We view the dihedral group of order eight as a unitriangular matrix group of degree three over field:F2, which is the group under multiplication of matrices of the form:

$$\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$$

with the entries over the field of two elements. We compare with the general theory of the conjugacy class structure of the group $$UT(3,q)$$, where $$q$$ is the field size. We denote by $$p$$ the prime number that is the characteristic of the field, so $$q$$ is a power of $$p$$.

Note that the letter $$a$$ used for matrix entries has no direct relation to the letter $$a$$ used for group elements of $$D_8$$.

Directed power graph
Below is a trimmed version of the directed power graph of the group. There is a dark edge from one vertex to another if the latter is the square of the former. A dashed edge means that the latter is an odd power of the former. We remove all the loops.