In joint work with Sílvia Anjos we continue the analysis by Abreu, MacDuff and Anjos of the topology of the group of symplectomorphisms of $S^2\times S^2$ when the ratio of the areas of the spheres lie in the interval $]1,2]$. We express the group of symplectomorphisms up to homotopy as the pushout (or amalgam) of certain compact Lie subgroups. We use this to give a homotopy decomposition of its classifying space and compute the corresponding ring of characteristic classes for symplectic fibrations.