Baby monster

In the area of modern algebra known as group theory, the baby monster group B (or, more simply, the baby monster) is a sporadic simple group of order

B is one of the 26 sporadic groups and has the second highest order of these, with the highest order being that of the monster group. The double cover of the baby monster is the centralizer of an element of order 2 in the monster group. The outer automorphism group is trivial and the Schur multiplier has order 2.

The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4. He investigated its properties and computed its character table. The first construction of the baby monster was later realized as a permutation group on 13 571 955 000 points using a computer by Jeffrey Leon and Charles Sims, though Robert Griess later found a computer-free construction using the fact that its double cover is contained in the monster. The name "baby monster" was suggested by John Horton Conway.

In characteristic 0 the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra, but Ryba (2007) showed that it does have such an invariant algebra structure if it is reduced modulo 2.

The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2.

Höhn (1996) constructed a vertex operator algebra acted on by the baby monster.

...
Wikipedia