= Euler characteristic from coset cycle counts
{title2=$\chi=n(2-2g_0-r)+\sum_{i=1}^r c_i$}
An $n$-sheeted surface cover of an orbifold of underlying <genus> $g_0$ with $r$ cone points has the displayed ordinary <Euler characteristic>, where $c_i$ is the number of cycles of the $i$th cone <monodromy permutation>. The ramification deficit over that point is $n-c_i$, so the formula follows by counting lifted vertices, edges and faces, or by the <Riemann-Hurwitz formula>. If every cone cycle has its full prescribed length, this equals $n\chi_{\mathrm{orb}}$.
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