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In this article we formulate a group of birational transformations which isisomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$with the aid of mutations and permutations of vertices to a mutation-periodicquiver on a torus. This group provides a class of higher order generalizationsof Jimbo-Sakai's $q$-Painlev\'e VI equation as translations on a root lattice.Then the known three systems are obtained again; the $q$-Garnier system, asimilarity reduction of the lattice $q$-UC hierarchy and a similarity reductionof the $q$-Drinfeld-Sokolov hierarchy.

Generalized $q$-Painlev\'e VI systems of type $(A_{2n+1}+A_1+A_1)^{(1)}$ arising from cluster algebra