  We investigate multi-agent epistemic modal logic with common knowledge
modalities for groups of agents and obtain van Benthem style model-theoretic
characterisations, in terms of bisimulation invariance of classical first-order
logic over the non-elementary classes of (finite or arbitrary) common knowledge
Kripke frames. The technical challenges posed by the reachability and
transitive closure features of the derived accessibility relations are dealt
with through passage to (finite) bisimilar coverings of epistemic frames by
Cayley graphs of permutation groups whose generators are associated with the
agents. Epistemic frame structure is here induced by an algebraic coset
structure. Cayley structures with specific acyclicity properties support a
locality analysis at different levels of granularity as induced by distance
measures w.r.t. various coalitions of agents.
