  We classify rational surfaces for which the image of the automorphisms group
in the group of linear transformations of the Picard group is the largest
possible. This answers a question raised by Arthur Coble in 1928, and can be
rephrased in terms of periodic orbits of a birational action of an infinite
Coxeter group on ordered point sets in the projective plane modulo projective
equivalence. We also outline the classification of non-rational surfaces with
large automorphism groups.
