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 $xy$?
There's a 12generator 80relator presentation for the Monster group. Specifically, we have 78 relators for the Coxeter group Y443:
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 nonidentity element in the centre of the group. See http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html for the explicit CoxeterDynkin diagram. 


A presentation for $M$ was constructed by A.A.Ivanov. This presentation is not so easy to describe; it arises from an amalgam of parabolics in certain diagram geometry. S.Norton then proved that the Ypresentation from ATLAS indeed describes the "Bimonster" (see Derek Holt's comment to the previous answer). 


Googling "monster group as a hurwitz group", the first hit is web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps which I believe contains the answer. 


Bimonster can be presented by the Coxeter group of the 26node graph of the projective plane of order 3: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.2634 (26 Implies The Bimonster, by John H. Conway and Christopher S. Simons). See also http://citeseer.uark.edu:8080/citeseerx/viewdoc/summary;jsessionid=57D5F004B2D42A5AE3CF28FD1C29E8E7?doi=10.1.1.137.1363 (An Elementary Approach to the Monster, by Christopher S. Simons). 

