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:
 $12$ relators of the form $x^2 = 1$, one for each node in the CoxeterDynkin 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 nonadjacent 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 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.

$\begingroup$ So, it is x^2=1 for each node, (xy)^3 for each pair of adjacent nodes, and [x,y]^2 ([x,y] is the commutator of x and y) for each pair of nonadjacent nodes. Then the "spider" relation, and quotienting out the center. Wow, that is very interesting. $\endgroup$ – Thomas Sep 16 '13 at 2:00

$\begingroup$ Also, what is the center element? It doesn't say on the page. $\endgroup$ – Thomas Sep 16 '13 at 2:08

$\begingroup$ What would happen if you did not include the spider relation? $\endgroup$ – Thomas Sep 16 '13 at 3:58

$\begingroup$ without the spider, you get some weird infinite Coxeter group. $\endgroup$ – Dima Pasechnik Sep 16 '13 at 6:54

1$\begingroup$ Interesting. What about the spider relation? How was that found? Were they intentionally trying to find the monster group, or was it an accident? $\endgroup$ – Thomas Sep 17 '13 at 11:12
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).

$\begingroup$ I never would have thought it would be the wreath product of the monster by Z2. The link you gave me does not let me see where the monster group is derived, let alone the whole chapter. Could someone email me a copy? $\endgroup$ – Thomas Sep 15 '13 at 14:03

$\begingroup$ sure, but where to? your profile doesn't reveal any email... $\endgroup$ – Dima Pasechnik Sep 15 '13 at 17:21

1$\begingroup$ dl.dropboxusercontent.com/u/11004520/CBO9780511629259A013.pdf $\endgroup$ – Dima Pasechnik Sep 15 '13 at 17:29
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).