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.
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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.
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2$\begingroup$ Alternatively, if you run the LowIndexSubgroups command (up to index 14 is enough) then you can quickly find quotients isomorphic to ${\rm PSL}(2,13)$ and $2^6.{\rm PSL}(2,7)$ as coset images of the subgroups of low index. $\endgroup$ Commented Sep 6, 2013 at 7:41
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$\begingroup$ Right, of course. It is interesting, there isn't a group 2^6.PSL(2,13) (at least as a quotient of the group). $\endgroup$– ThomasCommented Sep 6, 2013 at 7:46
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$\begingroup$ Out of curiosity, is there a quotient with composition series: PSL(2,13)-PSL(2,7)-2^3? $\endgroup$– ThomasCommented Sep 6, 2013 at 8:08
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1$\begingroup$ Yes there are two such quotients. The group is a direct product of ${\rm PSL}(2,13)$ with a group with structure $2^{3+3}.{\rm PSL}(2,7)$. $\endgroup$ Commented Sep 6, 2013 at 8:38
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$\begingroup$ Could you give a relation to add to make such a group? $\endgroup$– ThomasCommented Sep 6, 2013 at 8:40