= Solution
Write the <dihedral group> as $D_6=\langle r,s\mid r^6=s^2=1,\ srs=r^{-1}\rangle$, where $r$ rotates through $\pi/3$. There are six rotations and six reflections, so \b[the order is 12]. Conjugation by $s$ reverses a rotation, and conjugation by $r$ sends $r^js$ to $r^{j+2}s$. Consequently the <conjugacy classes> and their sizes are
$$
\boxed{\{1\}\ (1),\quad\{r^3\}\ (1),\quad\{r,r^5\}\ (2),\quad\{r^2,r^4\}\ (2),\quad\{s,r^2s,r^4s\}\ (3),\quad\{rs,r^3s,r^5s\}\ (3).}
$$
The last two classes distinguish the reflection axes through opposite vertices and through opposite edge midpoints. Their sizes sum to 12.
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