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We consider the distribution of the Galois groups$\operatorname{Gal}(K^{\operatorname{un}}/K)$ of maximal unramified extensionsas $K$ ranges over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. Weprove two properties of $\operatorname{Gal}(K^{\operatorname{un}}/K)$ comingfrom number theory, which we use as motivation to build a probabilitydistribution on profinite groups with these properties. In Part I, we buildsuch a distribution as a limit of distributions on $n$-generated profinitegroups. In Part II, we prove as $q\rightarrow\infty$, agreement of$\operatorname{Gal}(K^{\operatorname{un}}/K)$ as $K$ varies over totally real$\Gamma$-extensions of $\mathbb{F}_q(t)$ with our distribution from Part I, inthe moments that are relatively prime to $q(q-1)|\Gamma|$. In particular, weprove for every finite group $\Gamma$, in the $q\rightarrow\infty$ limit, theprime-to-$q(q-1)|\Gamma|$-moments of the distribution of class groups oftotally real $\Gamma$-extensions of $\mathbb{F}_q(t)$ agree with the predictionof the Cohen--Lenstra--Martinet heuristics.

A predicted distribution for Galois groups of maximal unramified extensions