1-transitive cyclic orderings

We give a classification of all the countable 1-transitive cyclic orderings, being those on which the automorphism group acts singly transitively. We also classify all the countable 1-transitive coloured cyclic orderings, where these are countable cyclic orderings in which each point is assigned a member of a set C, thought of as its ‘colour’, and by ‘1-transitivity’ we now mean that the automorphism group acts singly transitively on each set of points coloured by a fixed colour. We conclude by giving constructions of some uncountable cyclic orderings whose automorphism groups enjoy certain special properties.