In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$ , which is the intersection of a quartic and three general quadratics in $\mathbb {P}^5$ . We thus prove a stronger Bogomolov–Gieseker inequality for characters of stable vector bundles and stable objects on Calabi–Yau complete intersection $X_{2,4}$ . Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$ .