Properties
Some of the Niemeier lattices are related to sporadic simple groups. The Leech lattice is acted on by a double cover of the Conway group, and the lattices A124 and A212 are acted on by the Mathieu groups M24 and M12.
The Niemeier lattices, other than the Leech lattice, correspond to the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when two points of the Leech lattice are joined by no lines when they have distance, by 1 line if they have distance, and by a double line if they have distance .
Niemeier lattices also correspond to the 24 orbits of primitive norm zero vectors of the even unimodular Lorentzian lattice II25,1.
Read more about this topic: Niemeier Lattice
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