  Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence
over $G$, we mean a finite unordered sequence of terms from $G$, where
repetition is allowed, and we say that it is a product-one sequence if its
terms can be ordered such that their product equals the identity element of
$G$. The monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a
finitely generated C-monoid whence it has a finite commutative class semigroup.
It is well-known that the class semigroup is a group if and only if $G$ is
abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper we show
that the class semigroup is Clifford (i.e., a union of groups) if and only if
$|G'| \le 2$ if and only if $\mathcal B (G)$ is seminormal, and we study sets
of lengths in $\mathcal B (G)$.
