The Monster group is the largest of the sporadic simple groups, and has been proven by Wilson to also be a Hurwitz group. It has a presentation in terms of Coxeter groups, specifically Y443 along with the "spider" relator, and quotienting out by the center. However, I am interested in a presentation as a Hurwitz group.


Specifically, what is a presentation of the monster in terms of two elements a and b such that a has order 2, b has order 3, and ab has order 7? Also, what is the smallest possible order of the commutator of a and b in such a presentation?



Presentation of the Monster as a Hurwitz group
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>


