15.55 (2002)

a) The Monster, $M$, is a 6-transposition group. Pairs of Fischer transpositions generate 9 $M$-classes of dihedral groups. The order of the product of a pair is the coefficient of the highest root of affine type $E_8$. Similar properties hold for Baby $B$, and $F'_{24}$ with respect to $E_7$ and $E_6$ when the product is read modulo centres ($2.B, 3.F'_{24}$). Explain this. Editors’ Comment of 2005: Some progress was made in (C. H. Lam, H. Yamada, H. Yamauchi, Trans. Amer. Math. Soc., 355, no. 9 (2007), 4107–4123).
b) Note that the Schur multiplier of the sporadic simple groups $M, B, F'_{24}$ is the fundamental group of type $E_8, E_7, E_6$, respectively. Why?

See (J. McKay, in: Finite groups, Santa Cruz Conf. 1979 (Proc. Symp. Pure Math., 37), Amer. Math. Soc., Providence, RI, 1980, 183–186) and (R. E. Borcherds, Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. I (1998), 607–616).