Maximal unramified 3-extensions of imaginary quadratic fields and SL2(Z3)

The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL2(Z3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this was the case then it would imply that there is no upper bound on the possible lengths of a finite p-class tower.