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For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of closedorientable surface $S_g$ of genus $g$. In this paper, we derive a finitegenerating set for the liftable mapping class groups corresponding tofinite-sheeted regular branched cyclic covers of spheres. As an application, weprovide an algorithm to derive presentations of these liftable mapping classgroups, and the normalizers and centralizers of periodic mapping classescorresponding to these covers. Furthermore, we determine the isomorphismclasses of the normalizers of irreducible periodic mapping classes in$\mathrm{Mod}(S_g)$. Moreover, we derive presentations for the liftable mappingclass groups corresponding to covers induced by certain reducible periodicmapping classes. Consequently, we derive a presentation for the centralizer andnormalizer of a reducible periodic mapping class in $\mathrm{Mod}(S_g)$ of thehighest order $2g+2$. As final applications of our results, we recover thegenerating sets of the liftable mapping class groups of the hyperelliptic coverobtained by Birman-Hilden and the balanced superelliptic cover obtained byGhaswala-Winarski.

Generating the liftable mapping class groups of cyclic covers of spheres