Non--trivial homotopy in the contactomorphism group of the sphere.

The symmetries of the standard contact structure of a sphere generate families of contact structures. There exists a Serre fibration relating the space of contact structures and the group of contactomorphisms. The homotopy exact sequence for this fibration is studied and the non--triviality of certain elements in the homotopy groups of the contactomorphism group is concluded. Part of the argument applies to $3$--Sasakian manifolds due to their quaternionic symmetries. We comment on an alternative approach to the detection of non--triviality through the definition of a series of indices generalizing the Maslov index in the symplectic case.