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A conic of the Veronese surface in PG(5,3) is a quadrangle. If one suchquadrangle is replaced with its diagonal triangle, then one obtains a pointmodel $K$ for Witt's 5-$(12,6,1)$ design, the blocks being the hyperplanesections containing more than three (actually six) points of $K$. As such apoint model is projectively unique, the present construction yields an easycoordinate-free approach to some results obtained independently by H.S.M.Coxeter and G. Pellegrino, including a projective representation of the Mathieugroup $M_{12}$ in PG(5,3).

The Veronese Surface in PG(5,3) and Witt's 5-$(12,6,1$ Design