Since the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ seems to have resisted attacks of some powerful programs, I will turn to a group that seems to be a bit easier to analyse. The group

$$H := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19} \rangle$$

only has one of the simple groups that are a quotient of $G$, $J_1$. Therefore it seems it will be easier to analyse. Adding the relation $(abcbc)^i$ gives the trivial group for all $i$ less than 25 except for 15, where it gives $J_1$. What is the group

$$I := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19}, (abcbc)^{25} \rangle?$$

I have checked on magma using knuth bendix, and it takes too long, so magma stops the calculation.