Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always write elements of $D_n$ in disjoint cycle notation?
[Math] How to write dihedral group in cycle notation
abstract-algebracyclic-groupsdihedral-groupsgroup-theory
Best Answer
The dihedral group $D_n$ is a subgroup of $S_n$ for all $n\ge 3$, see also here, so we can write the elements in cycle notation. More precisely, $D_n$ is generated by an $n$-cycle $\sigma$ and a $2$-cycle $\tau$ satisfying the conditions $\sigma^n=\tau^2=1$, $\sigma\tau=\tau\sigma^{n-1}$. This gives a systematic way how to write the elements of $D_n$ in cycle notation. For example, for $n=3$, take $\sigma=(123)$ and $\tau=(23)$. Then $D_3=\{id,(12),(13),(23),(123),(132) \}$.