I have computed two pairs of generators $(a,b)$ of the Monster satisfying the relations $a^2 = b^3 = (ab)^7 = 1$ using [1]. In both cases $a$ is of class 2B, $b$ is of class 3B in the Monster, and the commutator $[a, b]$ has order 39. This gives an upper bound for the order of that commutator. More details of the computation are given in subdirectory applications/Hurwitz of [1].

[1] Martin Seysen, the python mmgroup package,

https://github.com/Martin-Seysen/mmgroup

I have computed two pairs of generators $(a,b)$ of the Monster satisfying the relations $a^2 = b^3 = (ab)^7 = 1$ using [1]. In both cases $a$ is of class 2B, $b$ is of class 3B in the Monster, and the commutator $[a, b]$ has order 39. This gives an upper bound for the minimal order of that commutator. More details of the computation are given in subdirectory applications/Hurwitz of [1].

[1] Martin Seysen, the python mmgroup package,

https://github.com/Martin-Seysen/mmgroup