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The $(p,q,r)$ triangle group is the group with presentation $[a,b | a^p=b^q=(ab)^r=1]$. Has anyone classified the (infinite) quotients of the $(2,4,4)$ triangle group?

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8  
The triangle group in question has a normal subgroup of index 8 isomorphic to $\mathbb{Z}^2$. Therefore, any quotient has a normal subgroup of index at most 8 that's a quotient of $\mathbb{Z}^2$. How much more information do you need? – HJRW Mar 4 '11 at 3:55
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So, in other words, yes. – Pete L. Clark Mar 4 '11 at 10:06
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That subgroup of index 8 is the commutator, so every quotient has a commutator which is (i) of index at most 8 and (ii) a quotient of $\mathbb{Z}^2$. – Steve D Mar 4 '11 at 16:51

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