An example of non-isomorphic group association schemes with the same parameters (Q1367601)

This paper gives an example of two finite groups \(G\) and \(H\), whose group association schemes \({\mathcal H}(G)\) and \({\mathcal H}(H)\) have the same parameters (i.e., \(G\) and \(H\) have the same character tables), but are not isomorphic. The respective groups \(G\) and \(H\) are two different (split and non-split) extensions of \({\mathbb{Z}}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\) by \(\text{SL}_3(2)\) (the group of non-singular \(3\times 3\)-matrices with coefficients in \(\mathbb{Z}_2\)). The fact that \({\mathcal H}(G)\not\simeq{\mathcal H}(H)\) is established by direct inspection of the structure of the groups using geometric arguments. Moreover, the automorphism groups of the association schemes are determined: \(\Aut({\mathcal H}(G))\simeq G\times G\) and \(\Aut({\mathcal H}(H))\simeq H\times H\). This gives another proof of the fact that \({\mathcal H}(G)\) and \({\mathcal H}(H)\) are non-isomorphic.