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Jun 7, 2015 at 20:27 comment added Alireza Abdollahi @YCor: I did not (couldn't maybe better to say) compute the kernel of $G\rightarrow \tilde{G}$. Instead as I mentioned in the question, I was unable to find the abelianization of $G^{(4)}$, where its index in $G$ is $19730006016$.
Jun 7, 2015 at 19:40 comment added YCor @AlirezaAbdollahi you could try to compute with GAP a presentation of the kernel of $G\to\tilde{G}$ and see if its abelianization is infinite? I don't remember if the index $\sim 10^5$ is within reasonable computing time.
Jun 7, 2015 at 15:49 comment added Alireza Abdollahi Many Thanks. It remains to decide if it is infinite or not.
Jun 7, 2015 at 15:47 vote accept Alireza Abdollahi
Jun 7, 2015 at 15:33 history answered Stefan Kohl CC BY-SA 3.0