The holomorphic symplectic automorphism group of a Z3-orbifold K3 is (Z3)^2 ⋊ Z4, realized inside M12 and M24, and it combines with Kummer symmetries to generate M24.
Not Doomed to Fail
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abstract
In their recent manuscript "An Uplifting Discussion of T-Duality", arXiv:1707.08888, J. Harvey and G. Moore have reevaluated a mod two condition appearing in asymmetric orbifold constructions as an obstruction to the description of certain symmetries of toroidal conformal field theories by means of automorphisms of the underlying charge lattice. The relevant "doomed to fail" condition determines whether or not such a lattice automorphism g may lift to a symmetry in the corresponding toroidal conformal field theory without introducing extra phases. If doomed to fail, then in some cases, the lift of g must have double the order of g. It is an interesting question, whether or not "geometric" symmetries are affected by these findings. In the present note, we answer this question in the negative, by means of elementary linear algebra: "geometric" symmetries of toroidal conformal field theories are not doomed to fail. Consequently, and in particular, the symmetry groups involved in symmetry surfing the moduli space of K3 theories do not differ from their lifts.
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Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups
The holomorphic symplectic automorphism group of a Z3-orbifold K3 is (Z3)^2 ⋊ Z4, realized inside M12 and M24, and it combines with Kummer symmetries to generate M24.