I want to check that this cycle notation is correct for the Dihedral Group of order $12$. I found this graph in Wikipedia. I did not find the cycle notation. May you please tell me if my cycle notation is correct?
Here is my solution: I will name the vertices $1,2,3,4,5,6$ in the clockwise direction.
1) Rotation 0 degrees
$(1)(2)(3)(4)(5)(6)$
2) Rotation 60 degrees (clockwise direction)
$(123456)$
3) Rotation 120 degrees
$(135)(246)$
4) Rotation 180 degrees
$(14)(25)(36)$
5) Rotation 240 degrees
$(153)(264)$
6) Rotation 300 degrees
$(165432)$
7) Flip 2-5
$(2)(5)(13)(46)$
8) Flip 1-4
$(1)(4)(26)(35)$
9) Flip 3-6
$(3)(6)(15)(24)$
10) Flip A
$(12)(36)(45)$
11) Flip B
$(14)(23)(56)$
12) Flip C
$(16)(25)(34)$
Best Answer
There is a computationally easier way to do this, and you do not even need to revisit the geometric intepretation of $\operatorname{Dih}(6)$. Let $\phi:\operatorname{Dih}(6)\times\{1,2,3,4,5,6\}\rightarrow\{1,2,3,4,5,6\}$ be the group action of the dihedral group of a regular hexagon on the six vertices. Let $\varphi:\operatorname{Dih}(6)\rightarrow\operatorname{Sym}(6)$ be the permutation representation afforded by the action. Then write $\varphi(r)=(123456)$ and $\varphi(s)=(13)(46)$. Then since $\varphi$ is a homomorphism, and $\operatorname{Dih}(6)=\{1,r,r^2,\dots,s,sr,sr^2\dots\}$ is generated by $r$ and $s$, you can compute the cycle representation of each element of $\operatorname{Dih}(6)$, e.g. $\varphi(sr^2)=\varphi(s)\varphi(r)^2=(13)(46)(123456)^2=\dots$ (check it yourself :D).
P.S. Hmm... it seems that your answer is correct, but you should check it again using the group action approach anyway.