I have been working with Hurwitz groups, and I came across the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, ([a,b]^2ab)^6 \rangle$. I'm trying to figure out exactly what this group is. I know it contains two copies of ${\rm PSL}(2,13)$, and there seems to be a very large 2-group in there as well, of order $2^{28}$ at least. Is this group even finite, and if so, what is the order? Does it contain any other simple groups?



What is this quotient of the triangle 2-3-7 group?
It is infinite.


As you pointed out yourself, the kernels of the maps onto ${\rm PSL}(2,13)$ have $2$-quotients of apparently unbounded order, which can be computed using the $p$-quotient algorithm.


I also found homomorphisms onto the Janko sporadic groups J1 and J2. Then I found a subgroup of index $525$ in J2 of which the inverse image in $G$ has infinite abelianization. Here are some Magma commands to do some of these computations.



```
> G<a,b> := Group<a,b|a^2,b^3,(a*b)^7,((a,b)^2*a*b)^6 >;
> SQ := SimpleQuotients(G,1000000: Limit:=10 );
> ChiefFactors(Image(SQ[1][1]));
    G
    |  A(1, 13)                   = L(2, 13)
    1
> P := pQuotient(Kernel(SQ[1][1]), 2, 3 : Print:=1);

Lower exponent-2 central series for $

Group: $ to lower exponent-2 central class 1 has order 2^14

Group: $ to lower exponent-2 central class 2 has order 2^42

Group: $ to lower exponent-2 central class 3 has order 2^70 

> ChiefFactors(Image(SQ[2][1]));
    G
    |  J1
    1
> h := SQ[3][1];
> I := Image(h);
> ChiefFactors(I);
    G
    |  J2
    1
> L := LowIndexSubgroups(I,1000);
> [Index(I,l): l in L];
[ 840, 560, 280, 525, 315, 100, 1 ]
> p := CosetAction(I,L[4]);
> S := sub< G | h*p >;
> Index(G,S);
525
> AbelianQuotientInvariants(S);
[ 6, 0 ]

```

