Just to summarize, here is all I know about the group $G = \langle x,y \mid x^2=y^3=(xy)^7=[x,y]^{10}=1>$ and its quotients.

It has (essentially unique) homomorphisms onto the simple groups ${\rm PSL}(2,41)$, $J_1$ and ${\rm HJ} = J_2$. There are no other finite simple quotients of order up to $2 \times 10^9$ and no others of type ${\rm PSL}(2,q)$. All direct products of these three groups, like $J_1 \times J_2$ or ${\rm PSL}(2,41) \times J_1 \times J_2$ are also quotients.

Computationally, we can also examine the kernels of the homomorphisms onto the three simple groups. (In principle we could study the kernels onto the direct products, but that would be much harder, because the images are too large.)

The abelianization of the kernel $K$ of the map onto ${\rm PSL}(2,41)$ is free abelian of rank $42$ (this proves that $G$ is infinite) and the associated rational module for the action of ${\rm PSL}(2,41)$ is irreducible. However, on reduction mod 2, this module has submodules of dimensions $1$ and $21$. We can also study the $p$-quotients of $K$ for primes $p$, and it appears to have class $2$ $p$-quotients of order $p^{495}$ for all $p$, so it looks as though there are many very large $p$-quotients!

The abelianization of the kernel of the map onto $J_1$ is elementary abelian of order $11^{14}$ and the action of $J_1$ on this quotient is irreducible. There is also a class $2$ quotient of order $11^{28}$ and a class $3$ quotient of order $11^{42}$, which suggests a pattern.

I have not managed to compute the abelianization of the kernel of $G$ onto $J_2$ yet, but it certainly involves a large elementary abelian 2-group. *Update*: it is elementary abelian of order $2^{41}$.

These computations work by using the Reidemeister-Schreier algorithm to compute a presentation of a subgroup of finite index in a given finitely presented group. This presentation is initially on the Schreier generators and, for a 2-generator group, the number of these is roughly equal to the index of the subgroup. The presentation can be simplified by eliminating redundant generators, but this tends to make the group relations longer, and so the larger the index of the subgroup, the more complex is the computed presentation. This limits our ability to perform computations with the subgroup, like computing its abelianization. (There are alternative approaches to computing subgroup presentations, but in my experience they all ultimately have similar limitations.)

*Added later*: It turns out that the group $G$ also has quotients isomorphic to (at least) one of the simple groups $G_2(p)$, $G_2(p^2)$, $G_2(p^4)$ for almost all primes $p$. These are finite quotients of the images of 7-dimensional representations over number fields of degree 4 over ${\mathbb Q}$, constructed by Plesken and Souvignier. See the discussion in Other Quotient of Hurwitz group

simplequotients. The kernel of the homomorphism onto ${\rm PSL}(2,41)$ has an infinite free abelian quotient of rank 42, so there are arbitrarily large finite quotients. – Derek Holt Sep 9 '13 at 13:41