We consider naturally generated subgroups En of Brn. On the geometric side we show that En is the bifurcation braid monodromy group of the family of plane polynomial coverings of degree 3. On the algebraic side there is the Hurwitz action of the braid group Brn on the n-fold Cartesian product Brn3 of Br3 = ha, b | aba = babi. The stabiliser of finite alternating sequences of its generators a, b is expected to be given by En. We are able to prove this conjecture in a few cases and apply our results to give characterisations in the most basic instances of the paths realised by degenerations in families of polynomials as defined by Donaldson.