  In this article we formulate a group of birational transformations which 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 $q$-Painlev\'e 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.
