  We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, \sigma
\oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area
1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} (
S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, \omega)$. So far, the
monotone case was studied by J. Evans and he proved that this group is
contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group
Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,\omega) $
of symplectomorphisms that act trivially on homology is always connected and
recently they also computed its fundamental group. We describe, in full detail,
the rational homotopy Lie algebra of this group.
  We show that some particular circle actions contained in the
symplectomorphism group generate its full topology. More precisely, they give
the generators of the homotopy graded Lie algebra of $\rm{Symp} (S^2 \times S^2
\#\, 3\overline{ \mathbb C\mathbb P}\,\!^2, \omega)$. Our study depends on
Karshon's classification of Hamiltonian circle actions and the inflation
technique introduced by Lalonde-McDuff. As an application, we deduce the rank
of the homotopy groups of $\rm{Symp}({\mathbb C\mathbb P}^2 \#\,
5\overline{\mathbb C\mathbb P}\,\!^2, \tilde \omega)$, in the case of small
blow-ups.
