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In this paper at first a new measure of stability of clustering solutions over different bootstrap samples of a data set is proposed. Thereafter in this paper, a multiobjective optimization based clustering technique is developed which optimizes both the measures of symmetry and stability simultaneously to automatically determine the appropriate number of clusters and the appropriate partitioning from data sets having symmetrical shaped clusters. The proposed algorithm utilizes a recently developed simulated annealing based multiobjective optimization technique, AMOSA, as the underlying optimization method. Here assignment of points to different clusters are done based on a recently developed point symmetry based distance rather than the Euclidean distance. Results on several artificial and real-life data sets show that the proposed technique is well-suited to detect the number of clusters from data sets having point symmetric clusters.