The Monster group (actually the bimonster) has a presentation as Y555. Y555 is the quotient of a coxeter group (the coxeter diagram is a central node with three "spokes" coming out of it with length 5) by the "spider relation": (ab1c1d1ab2c2d2ab3c3d3)^10. In this presentation, a is the generator in the center of the coxeter diagram, and bi, ci, di is the first three generators on a spoke (for i ranging from 1 to 3).
I was experimenting with the smaller group Y222, and I found that it had order 12597120 = 2^7*3^9*5. Unfortunately, Magma was unable to find the composition factors of G. What are the composition factors?