Exceptional Extended Dual Polar Spaces

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We address the classification problem of flag-transitive geometries with diagrams of the form where the leftmost edge symbolizes the geometry of vertices and edges of a complete graph ons+2 vertices and the residue of an element of the leftmost type is a finite thick classical dual polar space. These geometries are known asextended dual polar spaces. An extended dual polar space is called affine if it possesses a flag-transitive automorphism group which contains a normal subgroup acting regularly on the set of elements of the leftmost type. For a dual polar space D with three points per line there exists a unique 2-simply connected affine extension A(D) of D. We show that a flag-transitive extended dual polar space is either a quotient of A(D) for some D or isomorphic to one of 19 exceptional geometries whose full automorphism groups are isomorphic respectively to Sym8,U4(2).2,Sp6(2)×2,Sp6(2), 3·U4(3).22,U4(3).22,U5(2).2,McL.2,HS.2,Suz.2,Sp8(2), 3·Fi22.2,Fi22.2,Co2× 2,Co2,Fi24(s=4,t=2),Fi24(s=t=3),F1andFi23.