Recent studies (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) of six-dimensional supersymmetric gauge theories that are engineered by a class of toric Calabi-Yau threefolds $X_{N,M}$, have uncovered a vast web of dualities. In this paper we analyse consequences of these dualities from the perspective of the partition functions $mathcal{Z}_{N,M}$ (or the free energy $mathcal{F}_{N,M}$) of these theories. Focusing on the case $M=1$, we find that the latter is invariant under the group $mathbb{G}(N)times S_N$: here $S_N$ corresponds to the Weyl group of the largest gauge group that can be engineered from $X_{N,1}$ and $mathbb{G}(N)$ is a dihedral group, which acts in an intrinsically non-perturbative fashion and which is of infinite order for $Ngeq 4$. We give an explicit representation of $mathbb{G}(N)$ as a matrix group that is freely generated by two elements which act naturally on a specific basis of the Kahler moduli space of $X_{N,1}$. While we show the invariance of $mathcal{Z}_{N,1}$ under $mathbb{G}(N)times S_N$ in full generality, we provide explicit checks by series expansions of $mathcal{F}_{N,1}$ for a large number of examples. We also comment on the relation of $mathbb{G}(N)$ to the modular group that arises due to the geometry of $X_{N,1}$ as a double elliptic fibration, as well as T-duality of Little String Theories that are constructed from $X_{N,1}$.
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