A correlation between ${\rm PSU}\sb{4}\,(3)$, the Suzuki group, and the Conway group

Abstract

We shall use a six dimensional projective representation of $PS{U_4}(3)$ of order ${2^7}{3^6}5 \cdot 7$ to construct 12 and $24$-dimensional complex projective representations of the Suzuki and Conway groups, respectively, acting on the Leech lattice. The construction makes it easy to show that the Suzuki and Conway simple groups have outer automorphism groups of order two and one, respectively. Also, the simple Suzuki group contains $3 \cdot PS{U_4}(3) \cdot 2,{3^5} \cdot {M_{11}}$, and a group which is probably $PS{U_5}(2)$, where $A \cdot B$ denotes an extension of the group $A$ by the group $B$.