Sporadic (FischerGriess) Monster group M = F_{1}
Order = 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^{46}.3^{20}.5^{9}.7^{6}.11^{2}.13^{3}.17.19.23.29.31.41.47.59.71.
Mult = 1.
Out = 1.
Robert Wilson's ATLAS page for the FischerGriess Monster M is available here.
M: Length ????, 2generator, ??relator.
< x, y  x^{2} = y^{3} = (xy)^{29} = [x, y]^{?} = (xyxyxy^{1})^{??} = ... = 1 >
Remark: x and y are R.A.Wilson's standard generators for M.
Only the insane would try to find a presentation for M on its standard generators. It is better to go for one of the Ydiagram presentations.
M × 2: Length 400, 12generator, 79relator (Y_{443}).
The Coxeter group (generated by involutions a, b_{1}, b_{2}, b_{3}, c_{1}, c_{2}, c_{3}, d_{1}, d_{2}, d_{3}, e_{1} and e_{2}) such that:
e_{1}  d_{1}  c_{1}  b_{1}  a  b_{2}  c_{2}  d_{2}  e_{2}

b_{3}

c_{3}

d_{3}
along with (ab_{1}c_{1}ab_{2}c_{2}ab_{3}c_{3})^{10} = 1 (the Spider relation).
Remark: This group is also presentated by Y_{553} and Y_{543}, where the Y_{443} subdiagrams generate and satisfy the above presentation.
(M × M):2: Length 452, 13generator, 92relator (Y_{444}).
The Coxeter group (generated by involutions a, b_{1}, b_{2}, b_{3}, c_{1}, c_{2}, c_{3}, d_{1}, d_{2}, d_{3}, e_{1}, e_{2} and e_{3}) such that:
e_{1}  d_{1}  c_{1}  b_{1}  a  b_{2}  c_{2}  d_{2}  e_{2}

b_{3}

c_{3}

d_{3}

e_{3}
along with (ab_{1}c_{1}ab_{2}c_{2}ab_{3}c_{3})^{10} = 1 (the Spider relation).
Remark: This group is also presentated by Y_{555} Y_{554} and Y_{544}, where the Y_{444} subdiagrams generate and satisfy the above presentation.
Ydiagrams.
Must mention Conway, Norton, Ivanov and others ... who actually did the work (like conjecturing and proving the above presentations).
Warning: There is more than one definition of Y_{pqr} in existence.
Last updated 15^{th} September, 1997
John N. Bray