A geometry of rank 5 associated with \(PGO_ 5(3)\) (Q1317463)

The author constructs a thin, residually connected geometry \(\Gamma\) of rank 5, whose residues of type \(\{i,i+1\}\) are hexagons, and those of type \(\{i,i+2\}\) are triangles. \(\Gamma\) admits the group \(PGO_ 5(3)\) of all projective collineations stabilizing a non-degenerate quadric in \(PG(4,3)\) as a flag-transitive (and primitive) automorphism group. The geometry \(\Gamma\) is constructed in terms of the generalized quadrangle of order (3,3) which belongs to \(PGO_ 5(3)\).