The average size of 3-torsion in class groups of 2-extensions

We determine the average size of the $3$ -torsion in class groups of G -extensions of a number field when G is any transitive $2$ -group containing a transposition, for example D₄ . It follows from the Cohen–Lenstra–Martinet heuristics that the average size of the p -torsion in class groups of G -extensions of a number field is conjecturally finite for any G and most p (including p |G| ). Previously this conjecture had only been proven in the cases of G=S₂ with $p=3$ and G=S₃ with $p=2$ . We also show that the average $3$ -torsion in a certain relative class group for these G -extensions is as conjectured, proving new cases of the Cohen–Lenstra–Martinet heuristics. Our new method also works for many other permutation groups G that are not $2$ -groups.