For computations with coherent configurations
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Spreads in Strongly Regular Graphs
Designs, Codes and Cryptography - Special issue dedicated to Hanfried Lenz
Three-Class Association Schemes
Journal of Algebraic Combinatorics: An International Journal
Computing isomorphisms of association schemes and its application
Journal of Symbolic Computation - Special issue on computer algebra and mechanized reasoning: selected St. Andrews' ISSAC/Calculemus 2000 contributions
On quasi-thin association schemes
Journal of Combinatorial Theory Series A
Association schemes generated by a non-symmetric relation of valency 2
Discrete Mathematics - Algebraic and topological methods in graph theory
Construction of association schemes from difference sets
European Journal of Combinatorics
Pseudocyclic association schemes arising from the actions of PGL(2, 2m) and PΓL(2, 2m)
Journal of Combinatorial Theory Series A
A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs
Journal of Algebraic Combinatorics: An International Journal
Permutation group approach to association schemes
European Journal of Combinatorics
Links between two semisymmetric graphs on 112 vertices via association schemes
Journal of Symbolic Computation
Nomura algebras of nonsymmetric Hadamard models
Designs, Codes and Cryptography
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We consider a rank 112 coherent configuration S=AP(2) on 28 points with 7 fibers of size 4. We describe S both axiomatically and as a model arising via the regular action of E"8 on the set of all 2-element subsets of an 8-element set. Moreover, we prove that our model is the unique structure, up to isomorphism, which satisfies the established axioms. A most important feature of S is that its group AAut(S) of algebraic automorphisms contains as a non-normal subgroup of index 8 the subgroup induced by all color automorphisms of S. This leads to a new type of automorphism of S, which we call ''proper algebraic''. All homogeneous mergings of S are described by us with the aid of a computer. Here, special attention is paid to so-called ''algebraic mergings'', i.e., those which arise from suitable subgroups of AAut(S). As a result we are able to give a unified explanation of various association schemes on 28 points, including those of pseudocyclic and quasithin type, plus some of pseudotriangular type. Moreover, we provide computer-free proofs that these schemes are in fact attainable via appropriate mergings of classes from S. Another interesting phenomenon is the existence of many ''twins'', i.e., pairs of non-isomorphic association schemes which are algebraically isomorphic inside S. Notable examples of twins are the triangular graph T(8) paired with one of the Chang graphs, and the Mathon pseudocyclic scheme paired with the pseudocyclic scheme of Hollmann. In all, we decribe four pairs of twins and one set of triplets in rather great detail.