  This article discusses deviations from the Cohen-Lenstra heuristics when
roots of unity are present. In particular, we propose an explanation for the
discrepancy between the observed number of cyclic cubic fields whose 2-class
group is $C_2\times C_2$ and the number predicted by the Cohen-Lenstra
heuristics, in terms of an invariant living in a quotient of the Schur
multiplier group. We also show that, in some cases, the definition of the
invariant can be simplified greatly, and we compute the invariant when the
cubic field is ramified at exactly one prime, up to $10^8$.
