Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$

Abstract

An action of a group $G$ on an Enriques surface $S$ is called Mathieu if it acts on $H^0(2K_S)$ trivially and every element of order 2, 4 has Lefschetz number 4. A finite group $G$ has a Mathieu action on some Enriques surface if and only if it is isomorphic to a subgroup of the symmetric group $\mathfrak S_6$ of degree 6 and the order $|G|$ is not divisible by $2^4$. The 'if' part is proved constructing explicit actions of three groups $\mathfrak S_5, N_{72}$ and $H_{192}$ on polarized Enriques surfaces of degree 30, 18 and 6, respectively and using the result by Keum--Oguiso--Zhang for the alternating group $\mathfrak A_6$.