Large Cayley graphs of diameter 2 for degrees 18, 19, 20, 22 and 27

Five Cayley graphs of diameter 2 with degrees 18, 19, 20, 22 and 27 and orders 210, 240, 264, 320 and 468. Each is a Cayley graph of a metacyclic group Z_m x| Z_k with an inverse-closed connection set. Their orders exceed the Cayley-graph table of the degree-diameter problem on combinatoricswiki (200, 200, 210, 288 and 392; page last modified 2022-02-18, read 2026-10-26). They are not records for the general degree-diameter problem: larger graphs of diameter 2 with no larger maximum degree are known for all five degrees. The general table lists 307, 338 and 381 vertices for degrees 18, 19 and 20; the 381-vertex graph has maximum degree 20, below 22; and Brown's construction from the projective plane of order 25 gives 651 vertices with maximum degree 26, below 27. The archive contains each graph's group parameters and connection set, a table of results, and a standard-library Python verifier that checks the group axioms exhaustively, that the connection set is inverse-closed and excludes the identity, and that the diameter is exactly 2 by breadth-first search from every vertex. A second check was run by a separate AI agent session with its own code. Searches and verification were carried out with Claude (Anthropic) under the direction of Kerem Özdemir. Correction (2026-10-26): the comparison with general-graph records was clarified.