I have the following group presentation:
$G=\left\langle a,b,c\ |\ a^2,b^{11},c^2,(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},(ac)^3,(bc)^2\right\rangle$
Is $G$ finite?
GAP's Size(G) runs out of memory pretty quickly. No surprise there.
I also tried using the ideas in the answer here to look for homomorphisms onto certain small simple groups, with no luck. And I used LowIndexSubgroups() to look for subgroups up to index $30$ with no results.
It's also worth noting that:
$a$ and $b$ generate a subgroup isomorphic to $M_{11}$ (The Mathieu Group on $11$ points)
$a$ and $c$ generate a subgroup isomorphic to $D_6$ (The Dihedral group of size $6$)
$b$ and $c$ generate a subgroup isomorphic to $D_{22}$ (The Dihedral group of size $22$)