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Of the five exceptional groups, $\mathrm{E}_6$ is considered the mostattractive for unification due to the following reasons: (i) it contains both$\mathrm{Spin} (10) \times \mathrm{U}(1)$ and $\mathrm{SU} (3) \times\mathrm{SU}(3) \times \mathrm{SU}(3)$ as maximal subgroups, each of which admitembeddings of the Standard Model; (ii) uniquely among the exceptional groups,it admits complex representations; in particular, its 27 dimensionalfundamental representation accommodates one generation of left-handed fermionsunder the usual charge assignments; (iii) all of its representations areanomaly-free. In this master's thesis, written in the spirit of Baez andHuerta's "The Algebra of Grand Unified Theories", we rigorously show how an$\mathrm{E}_6$ grand unified theory is mathematically constructed. Our modestcontribution to the literature includes an explicit check that that$\mathbb{Z}_4$ kernel of the homomorphism $\mathrm{Spin} (10) \times\mathrm{U}(1) \to \mathrm{E}_6$ acts trivially on every fermion; we alsoformulate symmetry breaking, in particular the symmetry breaking of the exotic$\mathrm{E}_6$ fermions under $\mathrm{Spin} (10) \to \mathrm{SU}(5)$, using adifferent approach than the usual Dynkin diagrams: we explicitly embedded$\mathfrak{su}(5) \hookrightarrow \mathfrak{so}(10) \cong \mathfrak{spin} (10)$and solve the related eigenvalue problem. Phenomenological aspects of grandunified theories are also discussed.

A Mathematical Construction of an E6 Grand Unified Theory