  We describe a general model for the free energy function for a homogeneous
medium of mutually interacting molecules, based on the formalism for a biaxial
nematic liquid crystal set out by Katriel {\em et al.} (1986) in an influential
paper in {\em Liquid Crystals} {\bf 1} and subsequently called the KKLS
formalism. The free energy is expressed as the sum of an entropy term and an
interaction (Hamiltonian) term. Using the language of group representation
theory we identify the order parameters as averaged components of a linear
transformation, and characterise the full symmetry group of the entropy term in
the liquid crystal context as a wreath product $SO(3)\wr Z_2$. The
symmetry-breaking role of the Hamiltonian, pointed out by Katriel {\em et al.},
is here made explicit in terms of centre manifold reduction at bifurcation from
isotropy. We use tools and methods of equivariant singularity theory to reduce
the bifurcation study to that of a $D_3\,$-invariant function on ${\bf R}^2$,
ubiquitous in liquid crystal theory, and to describe the 'universal'
bifurcation geometry in terms of the superposition of a familiar swallowtail
controlling uniaxial equilibria and another less familiar surface controlling
biaxial equilibria. In principle this provides a template for {\em all} nematic
liquid crystal phase transitions close to isotropy, although further work is
needed to identify the absolute minima that are the critical points
representing stable phases.
