These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 6. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {lambda1} {lambda2} {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 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/1, 0/3, 0/6, 1/6, 1/2, 2/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,infinity) The half-space computation determines rationality. The supplemental half-space computation is not needed. These NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points because every loop multiplier of the mod 2 slope correspondence graph is at least 1 and the map is rational. NONTRIVIAL CYCLES 1/0 -> 0/1 -> 1/0 The slope function maps some slope to the nonslope. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 ends in either one of the above cycles or the nonslope. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<1,b*c,d*a,c,c^-1,1>(2,3)(4,5)", "b=<1,1,1,1,1,c>(2,3)(4,5)", "c=<1,d,1,1,1,1>(1,2)(3,4)(5,6)", "d=(1,2)(3,4)(5,6)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,b*c,d*a,c,c^-1,c>(2,3)(4,5)", "b=<1,1,1,1,1,1>(2,3)(4,5)", "c=(1,2)(3,4)(5,6)", "d=(1,2)(3,4)(5,6)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,2)(3,4)(5,6)", "b=<1,d,1,1,1,1>(1,2)(3,4)(5,6)", "c=<1,1,1,1,1,c>(2,3)(4,5)", "d=<1,d*a,b*c,c^-1,c,1>(2,3)(4,5)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,d,b*c,d*a,c,c^-1>(1,2)(3,4)(5,6)", "b=(1,2)(3,4)(5,6)", "c=<1,1,1,1,1,1>(2,3)(4,5)", "d=<1,d*a,b*c,c^-1,c,c>(2,3)(4,5)", "a*b*c*d");