Rigid ramification type ($\mathrm{Ni}(G;C_1,\dots ,C_r)/G$)

Fix ordered branch positions and conjugacy classes $C_i$ in a finite group $G$. The generating product-one tuples $(g_1,\dots ,g_r)\in \prod C_i$ form the inner Nielsen set. The type is rigid when this set is nonempty and a single simultaneous-conjugation orbit. Over $\mathbb{C}(T)$ this fixes the connected Galois cover up to isomorphism: tuples prescribe the monodromy epimorphism of the punctured sphere, and conjugate epimorphisms give isomorphic covers; the local completions at punctures are uniquely determined by their cyclic monodromy.