  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 SL_2(Z_3). 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
were the case then it would imply that there is no upper bound on the possible
lengths of a finite p-class tower.
