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Let 
$\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus 
$g \geq 1$, and let 
$\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover 
$p:S \to X$, where X is a hyperbolic surface. For 
$k \geq 2$, let 
$p_k: S_{k(g-1)+1} \to S_g$ be the standard k-sheeted regular cyclic cover. In this paper, we show that 
$\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalising subgroups in 
$\mathrm{Mod}(S_g)$, which are also maximal when k is prime. Furthermore, we derive explicit finite generating sets for 
$\mathrm{LMod}_{p_k}(S_g)$ for 
$g \geq 3$ and 
$k \geq 2$, and 
$\mathrm{LMod}_{p_2}(S_2)$. For 
$g \geq 2$, as an application of our main result, we also derive a generating set for 
$\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(\iota)$, where 
$C_{\mathrm{Mod}(S_g)}(\iota)$ is the centraliser of the hyperelliptic involution 
$\iota \in \mathrm{Mod}(S_g)$. Let 
$\mathcal{L}$ be the infinite ladder surface, and let 
$q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by 
$\langle h^{g-1} \rangle$ where h is the standard handle shift on 
$\mathcal{L}$. As a final application, we derive a finite generating set for 
$\mathrm{LMod}_{q_g}(S_g)$ for 
$g \geq 3$.
