  In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a
generic dynamical system with the symmetry of the group $\mathbf{Q}_8,$
constructed via a Cayley graph. While the Hopf bifurcation is similar to that
of a $\mathbf{D}_8$--equivariant system, our main result comes from analyzing
the system under weak coupling. We identify the conditions for heteroclinic
cycle between three equilibria in the three--dimensional fixed point subspace
of a certain isotropy subgroup of $\mathbf{Q}_8\times\mathbf{S}^1.$ We also
analyze the stability of the heteroclinic cycle.
