  Let $H$ be a Krull monoid with finite class group $G$ and suppose that every
class contains a prime divisor. If an element $a \in H$ has a factorization
$a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots, u_k \in
H$, then $k$ is called the length of the factorization and the set $\mathsf L
(a)$ of all possible factorization lengths is the set of lengths of $a$. It is
classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$
of all sets of lengths depends only on the class group $G$, and a standing
conjecture states that conversely the system $\mathcal L (H)$ is characteristic
for the class group. We verify the conjecture if the class group is isomorphic
to $C_n^r$ with $r,n \ge 2$ and $r \le \max \{2, (n+2)/6\}$. Indeed, let $H'$
be a further Krull monoid with class group $G'$ such that every class contains
a prime divisor and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove
that, if one of the groups $G$ and $G'$ is isomorphic to $C_n^r$ with $r,n$ as
above, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).
