Infinite class field tower with small root discriminant

We extend Schoof's theorem from cyclic case to any finite case and apply this to construct a class of Z/m Z Z/φ(n) Z extensions of Q, where m is either a power of $2$ or an odd integer, and n be any integer such that m divides n. As an application, we give some number fields with small root discriminant, having infinite p-class field tower when $p=3, 5, 7$.