On the Geometry of Baer Subplanes inPG(2,4)

Let C be an L3(4)-orbit of Baer subplanes in PG(2, 4). We define a graph Γ on C where two subplanes are called adjacent if their intersection contains only one point. We determine Γ and the action of L3(4) is analysed in detail. It is shown that Γ is the collinearity graph of the geometry (101) of Buekenhout. We also give a detailed construction of this geometry. Moreover, we prove that, ifL3 (4) or L3(4) : 21are automorphism groups, this geometry is determined up to isomorphism by its diagram.