  Finite symmetries abound in particle physics, from the weak doublets and
generation triplets to the baryon octet and many others. These are usually
studied by starting from a Lie group, and breaking the symmetry by choosing a
particular copy of the Weyl group. I investigate the possibility of instead
taking the finite symmetries as fundamental, and building the Lie groups from
them by means of a group algebra construction.
  Finite group algebras are the natural algebraic structures in which finite
symmetry groups, such as the symmetry group of three generations of elementary
fermions, are embedded in Lie groups, that are necessary for the formalism of
quantum field theory, including the gauge groups of the fundamental forces.
They are also the natural algebraic structures for describing representations
of groups, which are used for describing elementary particles and their quantum
properties. It is natural therefore to ask the question whether finite group
algebras can provide a formal underpinning for the standard model of particle
physics, and if so, whether this foundation can explain any aspects of the
model that are otherwise unexplained, such as the curious structure of the
combined gauge group, or the mixing angles between the different forces.
  In this paper I investigate the relationships between finite symmetry groups
and the gauge groups of each of the fundamental forces individually and in
combination, and show that the geometry of representations of the finite groups
can be used to predict accurate values for a number of the mixing angles in the
standard model, including the electro-weak mixing angle, one lepton mixing
angle, one quark mixing angle and one of the CP-violating phases.
