Desargues, doily, dualities and exceptional isomorphisms

We study various graphs associated with symmetric groups. This leads to a quick understanding of automorphisms, dualities and polarities both for the Desargues configuration and the smallest generalised quadrangle. The results are used to establish the existence of outer automorphisms of the symmetric group on six symbols, and the exceptional isomorphism between the alternating group on six symbols and the little projective group of the projective line over the field with nine elements. For the latter isomorphism, we present a new proof of uniqueness for the inversive plane of order 3.