A non-Schurian coherent configuration on 14 points exists

For a few decades the smallest known non-Schurian coherent configuration was the association scheme on 15 points, coming from a doubly regular tournament. Last year the second author, using a computer, enumerated all coherent configurations of order up to 15. A consequence of the enumeration is that all coherent configurations up to 13 points are Schurian and a unique non-Schurian rank 11 coherent configuration of order 14 exists. This coherent configuration has two fibers of sizes 6 and 8, and an automorphism group of order 24 isomorphic to SL(2, 3). We provide a computer free interpretation of this new object, relying on some simple interplay between group theoretical and combinatorial arguments.