  Of the five exceptional groups, $\mathrm{E}_6$ is considered the most
attractive 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 admit
embeddings of the Standard Model; (ii) uniquely among the exceptional groups,
it admits complex representations; in particular, its 27 dimensional
fundamental representation accommodates one generation of left-handed fermions
under the usual charge assignments; (iii) all of its representations are
anomaly-free. In this master's thesis, written in the spirit of Baez and
Huerta's "The Algebra of Grand Unified Theories", we rigorously show how an
$\mathrm{E}_6$ grand unified theory is mathematically constructed. Our modest
contribution 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 also
formulate symmetry breaking, in particular the symmetry breaking of the exotic
$\mathrm{E}_6$ fermions under $\mathrm{Spin} (10) \to \mathrm{SU}(5)$, using a
different 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 grand
unified theories are also discussed.
