by $\alpha_j$ or a $\gamma_i$, so condition (iv) in the definition of the data set ensures that the kernel of $\rho$ is torsion-free. Therefore, the orbifold covering $S \to {\operatorname{{\mathcal O}}}$ corresponding to the kernel is a manifold, and calculation of the Euler characteristic shows that $S = S_g$. Thus we obtain a $C_n$-action $t$ on $S_g$ with $r$ distinguished fixed points ${\operatorname{{\mathbb P}}}(t)$. We now construct ${\operatorname{{\mathbb O}}}_t$ from ${\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}}$ in the following manner. For each pair $(p,m_p) \in {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}}$, write $p = (a,b)$. If $a=0$, then choose $m_p$ orbits of size $n$ (if $t$ is a free action, this choice is trivial, but otherwise, such an orbit would always exist in a small neighbourhood around any fixed point of $t$). If $a\neq 0$, then there exists a cone point in ${\operatorname{{\mathcal O}}}$ of degree $b$, so there exists an orbit of $t$ of size $n/b$ in $S_g$. Once again, by considering a small neighbourhood of this orbit, we may choose $m_p$ distinct orbits $\{{\operatorname{{\mathbb O}}}_p^1, {\operatorname{{\mathbb O}}}_p^2, \ldots, {\operatorname{{\mathbb O}}}_p^{m_p}\}$ and set $${\operatorname{{\mathbb O}}}(t) := \bigsqcup_{(p,m_p) \in {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}}} \{{\operatorname{{\mathbb O}}}_p^1, {\operatorname{{\mathbb O}}}_p^2, \ldots, {\operatorname{{\mathbb O}}}_p^{m_p}\},$$ which in turn gives ${\operatorname{{\mathbb O}}}_t = 