Find a subgroup of $D_6$ where $D_6$ is the regular hexagon, with 12 symmetries. $$ D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\} $$ where $r^6 = e$. And the $r^n$ represent rotations, and $ar^n$ represent reflections.
When I do the Cayley table for D6, what is the product of $(ar)(ar)$ ? I would say it is $a^2(r^2)$, but that is not part of the symmetries.
In other words, what is $a$ squared? I know that $D_6$ is abelian, so $r^2$ times something. And does the value of $a$ depend on the value of $n$?
Thank you.