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We study generators of the fundamental group of the group ofsymplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbbC\mathbb P}\,\!^2, \omega)$ for some particular symplectic forms. It wasobserved by J. K\c{e}dra that there are many symplectic 4-manifolds $(M,\omega)$, where $M$ is neither rational nor ruled, that admit no circle actionand $\pi_1 (\mathrm{Ham} (M,\omega))$ is nontrivial. On the other hand, itfollows from previous results that the fundamental group of the group$\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,k\,\overline{\mathbb C\mathbbP}\,\!^2, \omega)$, of symplectomorphisms that act trivially on homology, with$k \leq 4$, is generated by circle actions on the manifold. We show that, forsome particular symplectic forms $\omega$, the set of all Hamiltonian circleactions generates a proper subgroup in $\pi_1(\mathrm{Symp}_h({\mathbb C\mathbbP}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, \omega)).$ Our work depends onDelzant classification of toric symplectic manifolds, Karshon's classificationof Hamiltonian $S^1$-spaces and the computation of Seidel elements of somecircle actions.

Loops in the fundamental group of $\mathrm{Symp} (\mathbb C\mathbb  P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2)$ which are not represented by  circle actions