Simplicity of infinite groups Sorry about asking so many questions, but I am a bit further on in my classification, and I am up to the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^7 \rangle$. It has no small quotients (up to 500000), and I suspect that it is simple. Is there a way to check for simplicity in infinite groups?

UPDATE (edit by YC)
The original question about the group $G$ has been answered below. Now I am up to the two groups $H := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^8 \rangle$ and
$I := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^9 \rangle$. I need to know if they are trivial, finite (the order would be good), or infinite.
 A: I think Mark Sapir's cautionary joke that magma is just returning a value of $sin(\pi)$ is actually surprisingly accurate. The "Order" function on a finitely presented group, denoted by magma as GrpFP, Magma returns positive integer if the group is computed to be finite, "Infinity" if the group is known to be infinite (e.g. a map to $\mathbb{Z}$ exists), and "0" when its certificates of infinite order cannot be established, coset enumeration exhausts memory, and magma can't determine the order of the group. 
http://magma.maths.usyd.edu.au/magma/handbook/text/773#8529
EDIT: The OP uses $[a,b]=a\cdot b^{-1} \cdot a^{-1} \cdot b$, my explanation below is $[a,b]=a\cdot b \cdot a^{-1} \cdot b^{-1}$. The computation of Order with the OP's definition returns a 0, as noted in the comments below. 
Having said all that, the magma computation I ran gave a different answer for $G$. Magma is saying $G$ is trivial. I used a machine at the University of Texas, which might have more available memory for coset enumeration than the online magma calculator. 
$
> G<a,b>:=Group<a,b|a^\wedge 2,b^\wedge 3,(a*b)^\wedge 7,(a*b*a^\wedge -1*b^\wedge -1)^\wedge 10,((a*b*a^\wedge -1*b^\wedge -1)^\wedge 4*b)^\wedge 7\
>;\\
> Order(G);\\                                                                    
1\\
$
However, the orders of $H$ and $I$ don't seem to be computable with the available memory, so they might be infinite and they might not be.
A: In fact your group is trivial. Here are two different computations with Magma, the first using coset enumeration over the subgroup $\langle ab \rangle$, and the second using the Knuth-Bendix completion algorithm.
 > G<a,b>:=Group<a,b|a^2,b^3,(a*b)^7,(a,b)^10, ((a,b)^4*b)^7 >;
 > H:=sub<G|a*b>;
 > time Index(G,H:CosetLimit:=100000000,Hard:=true); 
 1
 Time: 27.730

 > time  R := RWSGroup(G:MaxRelations:=100000, TidyInt:=1000);
 Time: 90.350
 > Order(R);
 1

