  Delsarte theory, more specifically the study of codes and designs in
association schemes, has proved invaluable in studying an increasing assortment
of association schemes in recent years. Tools motivated by the study of
error-correcting codes in the Hamming scheme and combinatorial $t$-designs in
the Johnson scheme apply equally well in association schemes with irrational
eigenvalues. We assume here that we have a commutative association scheme with
irrational eigenvalues and wish to study its Delsarte $T$-designs. We explore
when a $T$-design is also a $T'$-design where $T'\supseteq T$ is controlled by
the orbits of a Galois group related to the splitting field of the association
scheme. We then study Delsarte designs in the association schemes of finite
groups, with a detailed exploration of the dicyclic groups.
