A higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlevé equation with $\tilde{W}((A_{2N}⋊A_1{)}^{(1)}\times A_1^{(1)})$ symmetry

Recently, a birational representation of an extended affine Weyl group of $(A_{2N}⋊A_1{)}^{(1)}$-type, which gives a higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlevé equation, was obtained. In this paper, we extend it to a birational representation of an extended affine Weyl group of $(A_{2N}⋊A_1{)}^{(1)}\times A_1^{(1)}$-type. Moreover, we provide conjectures on periodic reductions from systems of P$\mathrm{\Delta }$Es with the CAC property to Painlevé type $q$-O$\mathrm{\Delta }$Es.