We study a new domain in $\mathbb C^4$, namely the hexablock $\mathbb{ H}$ that arises in connection with a special case of the $\mu$-synthesis problem in $M_2(\mathbb C)$. Previous attempts to study a few instances of the $\mu$-synthesis problem led to domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$. Throughout, the relations of $\mathbb{ H}$ with these three domains are explored. Several characterizations of the hexablock are established. We determine the distinguished boundary $b \mathbb{ H}$ of $\mathbb{ H}$ and obtain a subgroup of the automorphism group of $\mathbb{ H}$. The \textit{ rational $\overline{ \mathbb{ H}}$-inner functions}, i.e., the rational functions from the unit disc $\mathbb{ D}$ to $\mathbb{ H}$ that map the boundary of $\mathbb{ D}$ to $b\mathbb{ H}$ are characterized. We obtain a Schwarz lemma for $\mathbb{ H}$. Finally, it is shown that $\mathbb{ G}_2$, $\mathbb{ E}$, and $\mathbb{ P}$ are analytic retracts of $\mathbb{ H}$. By an application of the theory of the hexablock, we find alternative proofs to a few existing results on geometry and function theory of the pentablock.
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