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A finite group $G$ is an $n$-transposition group if there exists a union $D\subset G$ of conjugacy classes of involutions such that $\langle D \rangle = G$ and for all $a,b\in D$, the product $ab$ is of order at most $n$.

The almost simple $3$-transposition groups were classified by Bernd Fischer. Among the groups classified are the three sporadic simple Fischer groups. It is also well-known that the Baby Monster group is a $4$-transposition group and the Monster is a $6$-transposition group.

Have $n$-transposition groups been classified or investigated for any other $n$ (especially $n=4,5,6$)?

What are the values of $n$ for the other exceptional and sporadic simple groups?

Are there other important generalisations of the $3$-tranposition property? (From Aschbacher's book on $3$-transposition groups, I see that there is a generalisation due to Timmesfeld to a $\{3,4\}^+$-transposition property, which is related to root elements of groups of Lie type.)

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Aschbacher and Hall classified groups generated by a class elements of elements of order 3. I do not know what kind of generalizations are you looking for, but these references could be useful:

Aschbacher, Michael; Hall, Marshall, Jr. Groups generated by a class of elements of order $3$. Finite groups '72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972), pp. 12--18. North-Holland Math. Studies, Vol. 7, North-Holland Amsterdam, 1973. MR0360794 (50 #13241), link.

Aschbacher, Michael; Hall, Marshall, Jr. Groups generated by a class of elements of order $3$. J. Algebra 24 (1973), 591--612. MR0311765 (47 #327), link.

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