Dihedral group:D16

Definition by presentation
The dihedral group $$\! D_{16}$$, sometimes denoted $$D_8$$, also called the dihedral group of order sixteen or the dihedral group of degree eight or the dihedral group acting on eight elements, is a dihedral group defined by the presentation:

$$\langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

where $$e$$ is the identity element.

The element $$a$$ is termed a rotation or a generator of the cyclic piece and the element $$x$$ is termed a reflection.

Note that the notation $$\! D_8$$ is more commonly used to denote the dihedral group of order eight.

Geometric definition
The dihedral group $$D_{16}$$ (also called $$D_8$$) is defined as the group of all symmetries of the regular octagon. This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by $$a$$) and has eight reflections each being an involution: four reflections about lines joining midpoints of opposite sides, and four reflections about diagonals.

Definition as a permutation group
The group can be defined as the subgroup of the symmetric group on $$\{ 1,2,3,4,5,6,7,8 \}$$ generated by a cycle of length $$8$$ and a reflection:

$$\! \langle (1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5) \rangle$$

Upto conjugacy
There are seven conjugacy classes of elements:


 * 1) The identity element. (1)
 * 2) The rotation by $$\pi$$, which is given by $$a^4$$ in the presentation. (1)
 * 3) The rotations by $$\pm \pi/2$$, which are given by $$a^{\pm 2}$$ in the presentation. (2)
 * 4) The rotations by $$\pm \pi/4$$, which are given by $$a^{\pm 1}$$ in the presentation. (2)
 * 5) The rotations by $$\pm 3\pi/4$$, which are given by $$a^{\pm 3}$$ in the presentation. (2)
 * 6) The rotations about axes joining opposite vertices, given by $$x, a^2x, a^4x, a^6x$$ in the presentation. (4)
 * 7) The rotations about axes joining midpoints of opposite sides, given by $$ax, a^3x, a^5x, a^7x$$ in the presentation. (4)

Upto automorphism
Under the action of outer automorphisms, the conjugacy classes (4) and (5) combine, and the conjugacy classes (6) and (7) combine. Thus, there are five equivalence classes of sizes $$1,1,2,4,8$$.