Bicyclic commutator quotients with one non-elementary component

For any number field K with non-elementary 3-class group Cl3(K) ≃ C3e × C3, e ≥ 2, the punctured capitulation type κ(K) of K in its unramified cyclic cubic extensions Li, 1 ≤ i ≤ 4, is an orbit under the action of S3 × S3. By means of Artin’s reciprocity law, the arithmetical invariant κ(K) is translated to the punctured transfer kernel type κ(G2) of the automorphism group G2 = Gal(F3(K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G′ ≃ C3e × C3, 2 ≤ e ≤ 6, according to the algebraic invariant κ(G), admits conclusions concerning the length of the Hilbert 3-class field tower F3 (K) of imaginary quadratic number fields K.

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