I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other than that I don't know. If we have two generators a and b such that $a^2=b^3=(ab)^7=1$, what are the possible orders of $[a,b]$? I believe that the conjugacy classes are known, so if you are given the conjugacy classes of two elements x and y, can we determine the possible conjugacy classes of $xy$?



Presentation of the Monster Group
There's a 12-generator 80-relator presentation for the Monster group. Specifically, we have 78 relators for the Coxeter group Y443:


* $12$ relators of the form $x^2 = 1$, one for each node in the Coxeter-Dynkin diagram;
* $11$ relators of the form $(xy)^3 = 1$, one for each pair of adjacent nodes;
* $55$ relators of the form $(xy)^2 = 1$ (commutators), one for each pair of non-adjacent nodes;


together with a single 'spider' relator, $(a b\_1 c\_1 a b\_2 c\_2 a b\_3 c\_3)^{10} = 1$, which results in the group $M \times C\_2$. We can get rid of the $C\_2$ by quotienting out by an eightieth relation, $x = 1$, where $x$ is the unique non-identity element in the centre of the group.


See <http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html> for the explicit Coxeter-Dynkin diagram.


