On the Class Group of an Imaginary Cyclic Field of Conductor 8p and 2-power Degree

Let p = 2e+1q + 1 be an odd prime number with 2 ∤ q. Let K be the imaginary cyclic field of conductor p and degree 2e+1. We denote by F the imaginary quadratic subextension of the imaginary (2; 2)-extension K(√2)/K+ with F ≠ K. We determine the Galois module structure of the 2-part of the class group of F.