Classifying Geometries with Cayley

In 1961, J.Tits has described a way to define a geometry from a group and a collection of subgroups. A lot of interesting geometrical objects arising from this definition have been studied by many geometers. The problem of finding all the geometries associated to a given group has been solved by hand, only for very small groups. A partial classification of the geometries of the Hall-Janko group has been recently obtained by M.Hermand, with the help of CAYLEY. Here we present a set of CAYLEY programmes to classify all the primitive, firm, residually connected and flag-transitive geometries associated to a given group G . As an application, we give the results obtained for the group W (E6).