Presentation of the Monster Group 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 $xy$?
 A: 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 Y-presentation from ATLAS indeed describes the "Bimonster" (see Derek Holt's comment to the previous answer). 
A: 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.
A: Bimonster can be presented by the Coxeter group of the 26-node 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). 
