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For odd $n\ge 3$, the dihedral group $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The group commutator $[r,s]=r^{-2}$ generates the rotation subgroup, while the quotient by that subgroup is $C_2$. Hence the commutator subgroup is exactly $⟨r⟩$ and the abelianization is $C_2$. The only abelian quotients are $C_2$ and the trivial group.

A coloring bracelet also identifies colorings related by reflection, so it is an orbit under a dihedral group rather than only a cyclic group.

On $K=\mathbb{Q}(x,y)$, let $\sigma (x,y)=(y,-x)$ and $\tau (x,y)=(x,-y)$. They generate a dihedral group of order eight, andThe inclusion from right to left is immediate. In the other direction, $x^2,y^2$ solve a quadratic over the displayed field and adjoining their square roots has degree at most four; the resulting total degree is at most eight, while the automorphism-count bound for a finite field extension gives the reverse bound.

For the order-eight dihedral group generated by $r$ of order four and $s$ of order two with $srs=r^{-1}$, let $H=K=⟨s⟩$. There are exactly three double cosets: $H$, $H{r}^{2}H$, and $HrH$, of sizes two, two, and four. Their number does not divide eight. This differs from the number of ordinary cosets, which divides group order by Lagrange's theorem.

Up to group isomorphism, the groups of order eight are $C_8$, $C_4\times C_2$, $C_2^3$, $D_8$ and $Q_8$, with $D_8$ here denoting the eight-element dihedral group. An element of order eight gives the first case. If every nonidentity element has order two the group is an elementary abelian group. Otherwise, after excluding elements of order eight, a cyclic subgroup of order four has index two; an element outside it either commutes with its generator or inverts it. In the latter case its square is either the identity or the central involution, giving the dihedral group or quaternion group.

For the dihedral group of a regular $2n$-gon, $n\ge 2$, use $r^{2n}=s^2=1$ and $srs=r^{-1}$. Its involutions are the central half-turn $r^n$ and all reflections $s{r}^j$. The half-turn has singleton conjugacy class and normal closure $\{1,r^n\}$. Conjugating a reflection changes its exponent by an even integer or negates it, so its conjugacy class is $\{s{r}^{j+2k}:0\le k<n\}$. Products of consecutive reflections generate $r^2$, giving normal closure $⟨r^2,s{r}^j⟩$, of order $2n$ and index two. The two reflection classes are distinguished by exponent parity.

A group is metacyclic when it has a cyclic normal subgroup whose quotient is cyclic. Every dihedral group is metacyclic through its rotation subgroup.

Every nonfaithful finite-dimensional complex representation of the infinite dihedral group is a direct sum of subrepresentations of dimension at most two. Its kernel contains a nonzero power of the rotation, so the representation factors through a finite dihedral group. Maschke's theorem makes that quotient representation semisimple, and all irreducible complex representations of a finite dihedral group have dimension at most two.