  The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves
defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of
the knots forms an abelian group under the connected sum and it can be
classified by the additive Vassiliev invariant of order $\leq k-1$. We see that
the set of $C_k$-equivalence classes of the spatial $\theta$-curves forms a
group under the vertex connected sum and that if the group is abelian, then it
can be classified by the additive Vassiliev invariant of order $\leq k-1$.
However the group is not necessarily abelian. In fact, we show that it is
nonabelian for $k\geq 12$. As an easy consequence, we have the set of
$C_k$-equivalence classes of $m$-string links, which forms a group under the
composition, is nonabelian for $k\geq 12$ and $m\geq 2$.
