  I show how the symmetry-breaking of a recently proposed embedding of the
standard model of particle physics in $E_8$ can be explained in terms of the
representation theory of the binary tetrahedral group. This finite group
provides a link between various types of spin and isospin that can be exploited
to `explain' the chirality of the weak interaction, and the existence of three
generations of fermions. Two apparently small technical differences between the
finite group model and the standard model turn out to have profound
consequences for the ways in which the weak and strong forces create mass.
