Isotropic models of evolution with symmetries

We consider isotropic Markov models on (phylogenetic) trees whose models of evolution are symmetric, that is invariant with respect to a transitive group of permutations of letters whose evolution we consider. Transitivity of the action of the group of symmetries implies strong bounds on the space of parameters of such a model. A special consideration is given to groups of symmetries containing large abelian subgroups. We prove that only hyperbinary models have abelian groups of symmetries. Using GAP, a computer algebra program, we calculate a complete classification of symmetric isotropic models on d letters, where d ≤ 9.