Group theory of cyclic cubic number fields

Astonishing new discoveries with quartets and octets of cyclic cubic fields sharing a common conductor are presented. Four kinds of graphs describing cubic residue conditions among the prime divisors of the conductor enforce elementary bi- or tricyclic 3-class groups and either a metabelian 3-class field tower group of coclass at least two or a closed Andozhskii-Tsvetkov group of order 6561. In the latter situation, abelian type invariants of first and second order are required for the identification.