We give conjectures on the distribution of the Galois groups of the maximal unramified extensions of Galois $\Gamma$ number fields or function

fields for any finite group $\Gamma$ (for the part of the Galois group prime to the order of $\Gamma$ and the order of roots of unity in the base

field). We explain some results about these Galois groups that motivate us to build certain random groups whose distributions appear in our

conjectures. We give theorems in the function field case (as the size of the finite field goes to infinity) that support these new

conjectures. In particular, our distributions abelianize to the Cohen-Lenstra-Martinet distributions for class groups, and so our

function field theorems give support to (suitably modified) versions of the Cohen-Lenstra-Martinet heuristics. This talk includes joint with

with Yuan Liu and David Zureick-Brown.

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