  We study the gauging of a discrete $\mathbb{Z}_3$ symmetry in the
five-dimensional superconformal $T_N$ theories. We argue that this leads to an
infinite sequence of five-dimensional superconformal theories with either $E_6
\times SU(N)$ or $SU(3)\times SU(N)$ global symmetry group. In the $M$-theory
realisation of $T_N$ theories as residing at the origin in the Calabi-Yau
orbifolds ${\mathbb{C}^3 \over {\mathbb{Z}_N \times \mathbb{Z}_N}}$ we identify
the $\mathbb{Z}_3$ symmetry geometrically and the new theories arise from
$M$-theory on the non-Abelian orbifolds $({\mathbb{C}^3 \over {\mathbb{Z}_N
\times \mathbb{Z}_N}})/{\mathbb{Z}_3}$. On the other hand, in the $(p,q)$
5-brane web description in Type IIB theory, the symmetry combines the
$U$-duality symmetry with a rotation in space, defining a so-called $U$-fold
background, where the $E_6$ symmetry is manifest.
