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We study the gauging of a discrete $\mathbb{Z}_3$ symmetry in thefive-dimensional superconformal $T_N$ theories. We argue that this leads to aninfinite sequence of five-dimensional superconformal theories with either $E_6\times SU(N)$ or $SU(3)\times SU(N)$ global symmetry group. In the $M$-theoryrealisation of $T_N$ theories as residing at the origin in the Calabi-Yauorbifolds ${\mathbb{C}^3 \over {\mathbb{Z}_N \times \mathbb{Z}_N}}$ we identifythe $\mathbb{Z}_3$ 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 $U$-foldbackground, where the $E_6$ symmetry is manifest.

Gauging Discrete Symmetries of $T_N$-theories in Five Dimensions