Element structure of dihedral groups

This article discusses the element structure of the dihedral group $$D_{2n}$$ of degree $$n$$ and order $$2n$$, given by the presentation:

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

Here $$e$$ denotes the identity element.

Summary
The information below is for $$D_{2n}$$.

Odd degree case
This is the case $$n$$ is odd, so $$2n$$ is twice an odd number.

Conjugacy class structure
 

Order information
We have the following number of elements of various orders:

Equivalence classes up to automorphism
It turns out that the equivalence classes up to automorphism are given precisely by order of elements, i.e., two elements are automorphic elements if and only if they have the same order. Equivalently, two elements are automorphic if and only if they generate the same cyclic subgroup. More explicitly:

Even degree case
Suppose $$n = 2m$$, and $$D_{2n}$$ is the dihedral group of order $$2n$$. Then, $$D_{2n}$$ has the following conjugacy classes (a total of $$(n + 6)/2$$ conjugacy classes):

Order information
We have the following number of elements of various orders:

Equivalence classes up to automorphism
Note that the data in the table below is not correct for the case $$n = 2$$, in which case we get the Klein four-group.

The equivalence classes up to automorphism are as follows: