Finite classical groups and flag-transitive triplanes

Let DD be a finite nontrivial triplane, i.e. a 2-(v,k,3)(v,k,3) symmetric design, with a flag-transitive, point-primitive automorphism group GG. If GG is almost simple, with the simple socle XX of GG being a classical group, then DD is either the unique (11, 6, 3)-triplane, with G=PSL2(11)G=PSL2(11) and Gα=A5Gα=A5, or the unique (45, 12, 3)-triplane, with G=X:2=PSp4(3):2≅PSU4(2):2G=X:2=PSp4(3):2≅PSU4(2):2 and Gα=Xα:2=(2⋅(A4×A4).2):2, where αα is a point of DD.