  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 $K$ for Witt's 5-$(12,6,1)$ design, the blocks being the hyperplane
sections containing more than three (actually six) points of $K$. As such a
point model is projectively unique, the present construction yields an easy
coordinate-free approach to some results obtained independently by H.S.M.
Coxeter and G. Pellegrino, including a projective representation of the Mathieu
group $M_{12}$ in PG(5,3).
