Cohen–Lenstra–Martinet conjecture for class groups of Γ-extensions

Let $S$) be a finite set of primes not containing any divisor of $2|\Gamma|$, let $\Gamma_\infty\subset\Gamma$ have order $1$ or $2$, and let $\mathcal{F}(\Gamma_\infty,X)$ be the family of Galois extensions $K/\mathbf{Q}$ with group $\Gamma$, with $\Gamma_\infty$ as an archimedean decomposition group, and with product of ramified primes at most $X$. Write $\mathrm{Cl}^S_K=\mathrm{Cl}_K\otimes_{\mathbf{Z}}\mathbf{Z}_{(S)}$. For a finite $\mathbf{Z}_{(S)}[\Gamma]$-module $H$, assume $H^\Gamma$ is trivial. Cohen–Lenstra–Martinet conjecture. There is a real number $c$, depending on $\Gamma$, $\Gamma_\infty$, and $S$, such that the probability that $\mathrm{Cl}^S_K\cong H$ approaches

$$c|H^{\Gamma_\infty}|^{-1}|\operatorname{Aut}_\Gamma(H)|^{-1}$$

as $X\to\infty$. This is a formulation of the Cohen–Lenstra–Martinet heuristics for $\Gamma$-extensions, supported by geometric and function-field evidence; the asserted limiting distribution remains conjectural in this generality.