McKay’s E 6 Observation on the Largest Fischer Group

Abstract

In this paper, we study McKay’s E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra \({VF^{\natural}}\) of central charge \({23\frac{1}{5}}\) on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so-called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay’s E6-observation via the theory of vertex operator algebras.