BCF-groups with elevated rank distribution

Infinitely many large Schur σ-groups G with logarithmic order lo(G) = 19+ e, nonelementary bicyclic commutator quotient G/G′ ≃ C3e × C3, e ≥ 2, elevated rank distribution ̺(G) = (3, 3, 3; 3), punctured transfer kernel type κ(G) ∼ (144; 4) and soluble length sl(G) = 3 are constructed. Up to e ≤ 4, they are realized as 3-class field tower groups Gal(F3 (K)/K) of imaginary quadratic number fields K = Q( √ d), d < 0. Their metabelianizations M = G/G′′ are BCF-groups with lo(M) = 8+e and bicyclic third lower central factor γ3(M)/γ4(M) ≃ C3×C3.

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