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Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are finite-dimensional and their Hilbert series have a factorization into quantum integers. Also, all known finite-dimensional elementary Nichols algebras of group type turn out to have many cubic relations. The technical assumption of our theorem can be removed if a conjecturedoi:10.1090/s0002-9947-2015-06231-x fatcat:upumbbergbabbb7n5qwjtfmxmq