Finite classical groups and flag-transitive triplanes

Let D be a finite nontrivial triplane, i.e. a 2- ( v , k , 3 ) symmetric design, with a flag-transitive, point-primitive automorphism group G . If G is almost simple, with the simple socle X of G being a classical group, then D is either the unique (11, 6, 3)-triplane, with G = P S L 2 ( 11 ) and G α = A 5 , or the unique (45, 12, 3)-triplane, with G = X : 2 = P S p 4 ( 3 ) : 2 ≅ P S U 4 ( 2 ) : 2 and G α = X α : 2 = ( 2 ⋅ ( A 4 × A 4 ) . 2 ) : 2 , where α is a point of D .