\(\Lambda_{\mathfrak{G}_{80}}(-1)\)

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. The coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) is negative definite of rank 19 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the groups \(\mathfrak{G}_{80} \cong F_{384}\), \(\mathfrak{G}_{28} \cong \Gamma_2 d\), \(\mathfrak{G}_{29} \cong Q_{16}\), \(\mathfrak{G}_{43} \cong \Gamma_7 a_2\), \(\mathfrak{G}_{44} \cong \Gamma_3 e\), \(\mathfrak{G}_{45} \cong \Gamma_6 a_2\), \(\mathfrak{G}_{58} \cong \Gamma_{22} a_1\), \(\mathfrak{G}_{59} \cong \Gamma_{23} a_2\), \(\mathfrak{G}_{60} \cong \Gamma_{26} a_2\), \(\mathfrak{G}_{67} \cong 4^2 D_6\), \(\mathfrak{G}_{71} \cong F_{128}\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 80 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[5,6] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5 that is number 6 of rank 5 in their Table 1. It satisfies their K3 criterion \(\operatorname{rk} B \geq 5\), \(\operatorname{rk} B - \operatorname{rk} A_B \geq 2\), and among the coinvariant lattices that satisfy it, it is the only one whose rank and discriminant form are those of \(\Lambda_G(-1)\): its discriminant form is the form \(q\) of row 80 of Table 10.3. The sum of its twist by \(-1\) and \(\Lambda^G\) therefore glues to an even unimodular lattice of signature \((3, 19)\), on which the stabilizer of \(B\) in \(\mathrm{Co}_0\) acts with this coinvariant lattice.