A construction of the Shephard–Todd group $G_{32}$ through the Weyl group of type ${\mathrm{E}}_6$

Abstract

It is well known that the quotient of the derived subgroup of the Shephard–Todd complex reflection group $G_{32}$ (which has rank $4$) by its center is isomorphic to the derived subgroup of the Weyl group of type ${\mathrm{E}}_6$. We show that this isomorphism can be realized through the second exterior power, and take the opportunity to propose an alternative construction of the group $G_{32}$.