© 2019 Elsevier B.V. In this paper we consider a two-parameter quadratic three-dimensional system with only six terms and two nonlinearities. First we analyze the Hopf bifurcation of its only equilibrium detecting several degeneracies. With this information we numerically obtain various bifurcation diagrams of periodic orbits in which saddle-node and period-doubling bifurcations as well as homoclinic connections appear. A careful study of the homoclinic orbits, in a region of the two-parameter plane where the equilibrium is a real saddle, shows the presence of a homoclinic flip bifurcation of case C. Here this orbit changes from orientable to non-orientable, being the lowest codimension for a homoclinic bifurcation to a real saddle equilibrium that results in chaotic behavior. More specifically, we determine that it corresponds to the case inward twist C in , in such a way that, as far as we know, it is the first example of a 3D vector field exhibiting this case.