Now that we have found all groups with a

^{2}=b

^{3}=(ab)

^{6}=1, the classification is not too hard, all groups in this class are a direct product of two cyclic groups with a semidirect product by Z

_{6}. The two cyclic groups have a ratio equal to squarefree products of 3 and primes that are 1(mod 3). It is now time to move on to <a,b|a

^{2}=b

^{3}=(ab)

^{7}>. Apparently this is where is gets really interesting.

Wikipedia wrote:All 26 sporadic groups are quotients of triangle groups, of which 12 are Hurwitz groups (quotients of the (2,3,7) group).

Well that's interesting. Seems like this will be a very hard case to cover. Also, there is a link to a paper that seems to say that the monster group is a (2, 3, 7) group. That is a group of order 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000! This is insane!

Also, I entered the groups <a,b|a

^{2}=b

^{3}=(ab)

^{7}=[a,b]

^{n}> into magma for n ranging from 2 to 7, and I got some very interesting results. First, forcing the order of [a,b] to be 2, 3, or 5 collapses the group to the trivial group. Second, whether [a,b] has order 6 or 7, they are both the group <1092,25> (why is there a factor of 13 in this group order?, I want to know an element that has order 13). If the order of [a,b] is 4, then we get the group <168,42> which I strongly suspect is the simple group of order 168. If [a,b] has order 8, then it is a group of order 10752, which magma is unable to classify. The groups seem to be growing a lot faster than in the (ab)

^{6}=1 case. I suspect that if the order of [a,b] is high enough, the group will be infinite. In fact, if [a,b] has order 9, then magma is unable to identify the group, so it could even be infinite for n=9!

Edit: I was right! I checked on a different site, and <168,42> is indeed the simple group of order 168. Fortunately, that means that there are no quotients, as there are no normal subgroups other than the trivial group and the whole group. Also, the group of order 1092 is simple as well! That is interesting.

Edit 2: It occurs to me that we can still generate the groups with [a,b], [a,b

^{2}], and ab, so we shall still refer to them as x, y, and z respectively. They might come in handy, especially since we know that the order of [a,b] is the same as the order of [a,b

^{2}], and ab has order 7. I wonder what the presentation in terms of this is (what y

^{x}, z

^{x}, and z

^{y} are).

Edit 3: I strongly suspect that the group of order 10752 has only the group of order 168 as a quotient.

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.