  The fundamental group of the complement of a plane curve is a very important
topological invariant. In particular, it is interesting to find out whether
this group is determined by the combinatorics of the curve or not, and whether
it is a direct sum of free groups and a free abelian group, or it has a
conjugation-free geometric presentation.
  In this paper, we investigate the structure of this fundamental group when
the graph of the conic-line arrangement is a unique cycle of length $n$ and the
conic passes through all the multiple points of the cycle. We show that if n is
odd, then the affine fundamental group is abelian but not conjugation-free. For
the even case, if n>4, then using quotients of the lower central series, we
show that the fundamental group is not even a direct sum of a free abelian
group and free groups.
