c-Extensions of the F4(2)-building Original Research Article

We construct four geometries View the MathML source with the diagram such that any two elements of type 1 are incident to at most one common element of type 2 and three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The automorphism group Ei of View the MathML source is flag-transitive, isomorphic to View the MathML source, View the MathML source, View the MathML source and View the MathML source, for i=1,2,3 and 4. We calculate the suborbit diagram of the collinearity graph of View the MathML source with respect to the action of Ei. By considering the elements in View the MathML source fixed by a subgroup Ti of order 3 in Ei we obtain four geometries View the MathML source with the diagram on which CEi(Ti) induces flag-transitive action, isomorphic to View the MathML source, View the MathML source, View the MathML source and View the MathML source for i=1,2,3 and 4. Next, by considering the elements fixed by a subgroup Si of order 7 in Ei we obtain four geometries with the diagram on which CEi(Si) induces flag-transitive action isomorphic to View the MathML source, View the MathML source, View the MathML source and View the MathML source, for i=1,2,3 and 4.