Central configurations of the n-body problem play an important role in the study of celestial mechanics. In this paper, we study six-body planar central configurations having certain symmetries. Special attention is given to the existence results of such configurations. With analytic proofs, we show the existence of a new central configuration, which is convex but not strictly convex, a non-symmetric concave case of central configuration, and some cases of stacked central configurations, which are central configurations with subsets of the bodies also forming central configurations.

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