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We construct a pair of non-commutative rank 8 association schemes from a rank3 non-symmetric association scheme. For the pair, two association schemes havethe same character table but different Frobenius-Schur indicators. Thissituation is similar to the pair of the dihedral group and the quaternion groupof order 8. We also determine the structures of adjacency algebras of them overthe rational number field.

A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes