c-Extensions of the F4(2)-building - Ivanov, A. A. (2012)
We construct four geometries ε1,..., ε4 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 εi is flag-transitive, isomorphic to 2E6(2): 2,3.2 E6(2) :2,226: F4(2) and E6(2) : 2, for i=1,2,3 and 4. We calculate the suborbit diagram of the collinearity graph of εi with respect to the action of Ei. By considering the elements in εi fixed by a subgroup Ti of order 3 in Ei we obtain four geometries T1,...,T4 with the diagram on which CEi(Ti) induces flag-transitive action, isomorphic to U6(2): 2,3. U6(2): 2, 214: Sp6(2) and L6(2): 2 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 L3(4): 2, 3.L3(4): 2, 28: L3(2) and (L3(2)xL3(2)): 2, for i=1,2,3 and 4.