Automorphisms of Niemeier lattices for Miyamoto's $${\mathbb {Z}}_3$$ -orbifold construction.

We classify, up to conjugation, all automorphisms of Niemeier lattices to which we can apply Miyamoto's orbifold construction. Using this classification, we prove that the VOAs obtained in Miyamoto (Symmetries, integrable systems and representations. Springer proceedings in mathematics and statistics, vol 40. Springer, London, pp 319-344, ) and Sagaki and Shimakura (Trans Am Math Soc, to appear) are all of holomorphic non-lattice VOAs which we can obtain by applying the $${\mathbb {Z}}_3$$ -orbifold construction to a Niemeier lattice and its automorphism. [ABSTRACT FROM AUTHOR]