  In this paper, we prove that the moduli space $\overline{M}_{X}(\nu)$ of
$H$-Gieseker semistable sheaves on a smooth cubic threefold $X$ with Chern
character $\nu=(4,-H,-\frac{5}{6}H^{2},\frac{1}{6}H^{3})$ is non-empty, smooth
and irreducible of dimension $8$.
