First, recall that $D_n$ is the dihedral group of order $2n$, which consists of the symmetries of a regular $n$-gon. The elements of $D_n$ can be represented as $\{r^k, r^k f : 0 \leq k \leq n-1\}$, where $r$ is the rotation by $2\pi/n$ radians and $f$ is a
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