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We classify Nichols algebras of irreducible Yetter-Drinfeld modules overgroups such that the underlying rack is braided and the homogeneous componentof degree three of the Nichols algebra satisfies a given inequality. Thisassumption turns out to be equivalent to a factorization assumption on theHilbert series. Besides the known Nichols algebras we obtain a new example. Ourmethod is based on a combinatorial invariant of the Hurwitz orbits with respectto the action of the braid group on three strands.

Braided racks, Hurwitz actions and Nichols algebras with many cubic relations