The hexablock is a domain arising from a special case of the $\mu$-synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is called a hexablock-contraction or simply $\mathbb H$-contraction. We characterize the unitaries and isometries associated with $\mathbb H$-contractions. Two different types of dilation results for $\mathbb H$-contractions are obtained. We find connection of this theory with the operators associated with the symmetrized bidisc and tetrablock, two other domains related to the $\mu$-synthesis problem.
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