These Thurston maps are NET maps for every choice of translation term. They have degree 8. They are imprimitive, each factoring as a Thurston map with degree 4 followed by a Euclidean NET map with degree 2. The non-Euclidean factor has fewer than four postcritical points for every translation term, and so is not a NET map. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0,lambda2} {lambda1,lambda1+lambda2} Since no Thurston multiplier is 1, this modular group Hurwitz class contains only finitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 6. The pullback map is constant: every curve has a trivial preimage. The image of the pullback map lies in the geodesic (0/1,2/1). Every NET map in this modular group Hurwitz class is rational. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<1,c^-1,b^-1*a*b,1,1,c*d,1,1>(2,6)(3,7)", "b=(1,7)(2,8)(3,5)(4,6)", "c=(1,2)(3,4)(5,6)(7,8)", "d=(1,8)(2,3)(4,5)(6,7)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,a*b,1,c^-1,1,c,1,c*d>(1,3)(2,8)(4,6)(5,7)", "b=<1,1,a*b,c^-1,1,1,b^-1,c*d>(3,7)(4,8)", "c=(1,2)(3,4)(5,6)(7,8)", "d=(1,8)(2,3)(4,5)(6,7)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,4)(2,7)(3,6)(5,8)", "b=(1,2)(3,4)(5,6)(7,8)", "c=(1,7)(2,8)(3,5)(4,6)", "d=(1,5)(4,8)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,4)(2,7)(3,6)(5,8)", "b=(1,2)(3,4)(5,6)(7,8)", "c=<1,1,a*b,c^-1,1,1,b^-1,c*d>(3,7)(4,8)", "d=(1,7)(2,4)(3,5)(6,8)", "a*b*c*d");