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Let $D_{n}=\langle a,b \ | a^n=b^2=abab=e\rangle$ be a dihedral group. Assume that $b$ acts on $\mathbb{P}^1$ by $z\mapsto \overline{z}$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$. Assume also that $a$ acts on $\mathbb{P}^1$ "holomorphically". What action of $D_{2n}$ on $\mathbb{P}^1$ is possible?

I am aware of the action $a:z\rightarrow e^{\frac{2\pi i k}{n}}z$ for $1\le k \le n$. This yields non-faithful action of $D_{2n}$, as the action of $a$ and $b$ commute. Are there any other actions?

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    That's why the questioner says the action is non-faithful, I guess. It seems to me that there is no faithful action with your assumption.2012-09-18

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You've already assumed how $b$ acts on $\Bbb P^1$, so I'd guess you meant that "$a$ takes $z\mapsto e^{\frac{2\pi i}n}z$" in your last paragraph. What if $m$ is an integer coprime to $n$ and $a$ takes $z\mapsto e^{\frac{2\pi i m}n}z$?

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    Thank you for the reply, Cameron. I corrected typos. Yes, your actions also provide examples (which I essentially think the same as mine).2012-09-18