$p$-class groups of certain extensions of degree $p$

Let $p$ be an odd prime number. In this article we study the distribution of $p$-class groups of cyclic number fields of degree $p$, and of cyclic extensions of degree $p$ of an imaginary quadratic field whose class number is coprime to $p$. We formulate a heuristic principle predicting the distribution of the $p$-class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to-$p$-part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.