The Veronese surface in \(PG(5,3)\) and Witt's \(5-(12,6,1)\) design (Q1268729)

A conic of the Veronese surface in PG(5,3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model \({\mathcal H}\) for Witt's 5--(12,6,1) design, the blocks being the hyperplane sections containing more than three (actually six) points of \({\mathcal H}\). As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some result obtained independently by Coxeter and Pellegrino, including a projective representation of the Mathieu group \(M_{12}\) in PG(5,3). \(\copyright\) Academic Press.