Generalized q-Painleve VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

In this article we formulate a group of birational transformations that is isomorphic 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-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo-Sakai's q-Painleve VI equation as translations on a root lattice. Then the known three systems are obtained again: the q-Garnier system, a similarity reduction of the lattice q-UC hierarchy, and a similarity reduction of the q-Drinfeld-Sokolov hierarchy.