Under the standard matrix realization of the unit quaternions as
\(\mathrm {SU}(2)\), the concrete group \(T\) becomes a finite subgroup of
\(\mathrm {SU}(2)\). Let \(\tau _3\) be its faithful 2-dimensional complex
representation. The McKay graph has one vertex for each irreducible
complex representation \(\tau _i\); the number of edges from \(\tau _i\) to
\(\tau _j\) is the multiplicity of \(\tau _j\) in
\[
\tau _3\otimes \tau _i.
\]
For the binary tetrahedral group this graph is the affine Dynkin diagram
\(\widetilde {E}_6\). The character table, the seven irreducible
representations, and the tensor-product calculation are given in
[stekolshchik2008notes, table A.12, prop. A.10, and ex. A.11, pp. 172--174].
This graph connects the representation theory of \(2T\) with quivers
and the ADE classification. The same source relates a binary polyhedral
subgroup \(G\leq \mathrm {SU}(2)\) to the quotient
\(\mathbb {C}^2/G\), its invariant algebra, and its Kleinian singularity in
[stekolshchik2008notes, secs. A.3--A.4, pp. 158--161]. These are
routes toward quotient geometry and related orbifold constructions. They
are not consequences of the semidirect-product theorem alone.
Neither route assigns physical meaning to \(Q\), \(C\), an algebra
block, or a vertex of \(\widetilde {E}_6\). They supply mathematical
structures on which a physical correspondence could be stated and
tested. A proposed correspondence must still identify the action,
representation, preserved structure, and physical interpretation.