I was experimenting with various presentations for groups, and I stumbled upon $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [aba,b]^6 \rangle$. I found that it has order 11741184, but the magma calculator won't give me much more than that. What I would like to know is: What are the composition factors of this group? I know that PSL(2,7) is one of them (making the order of [aba,b] 3 instead of 6 gives PSL(2,7)), but other than that I'm not sure.



Help understanding a group
Actually, I just figured it out. The groups 
$H := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{28}, [aba,b]^6 \rangle$ and
$I := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{8}, [aba,b]^6 \rangle$ are quotients of this group, and H has composition series: PSL(2,7)-PSL(2,13), and the second has composition series PSL(2,7)-Z2^6. Therefore the whole group has composition series: PSL(2,13)-PSL(2,7)-Z2^6.


