The wreath product of the Monster group with Z2 (M ≀ Z2) is the BiMonster, a quotient of the Y_{555} Dynkin diagram which contains six E8(Y_{124}) Dynkin diagrams and the 23rd Niemeier lattice E8^{3} root system. This can be reduced to Y_{444} by applying the spider relation to E8^{3} .

Happy Family genarations:

Red=Mathieu groups (1st)

Green=Leech lattice groups (2nd)

Blue=other Monster subquotients (3rd)

Black=Pariahs

Each color coded node represents one of the 24 Niemeier lattices, and the lines joining them represent the 24-dimensional odd unimodular lattices with no norm 1 vectors. The Coxeter number of the Niemeier lattice is to the left. The red node index number indicates the row in the table above.

”The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that ”the exceptional Lie groups all exist because of the octonions: G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous

vector space).”

The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.