This may not be a research-level question, which is why I [submitted it to math.stackexchange](https://math.stackexchange.com/questions/953186) first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it here as well is poor etiquette.




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Let $G = PSL(2,q)$. I am interested in the different ways of writing $G$ as an image of the modular group. If $A, B \in G$ are elements of order $2, 3$ respectively, then $\langle A,B \rangle$ is an image of the modular group (and is quite often the whole group $G$). Call $(A,B)$ a $(2,3)$-pair, and say that the pairs $(A,B)$, $(A',B')$ are equivalent if there is an automorphism of $G$ taking one pair to the other.


For example, I looked at the case $q = 5$ and there was exactly one equivalence class of $(2,3)$-pairs for each possible order $2, 3, 5$ of the element $AB$ (unless I made a mistake).



> 
> Is this true in general? If not, is there a nice way to list the equivalence classes in $G$?
> 
> 
> 


My motivation for asking about this is [here](https://mathoverflow.net/questions/180011/connected-cayley-graphs-of-psl2-q-from-2-3-n-triples).



Equivalence classes of (2,3)-pairs in PSL(2,q)
Concerning the discussion for previous answer (I'm not allowed to comment yet), the discrepancy comes between counting (2,3,5)-subgroup generator tuples, or subgroups of type A5:


A5 has two non-conjugate (2,3,5)-triples (that are conjugate under S5), so it is plausible that there are two orbits of generator triples and one orbit of subgroups.


Concerning the original question, this gets quickly more complicated for larger $q$: The GAP code:



```
f:=FreeGroup("x","y");
f:=f/ParseRelators(f,"x^2,y^3");
GQuotients(f,PSL(2,7)); # replace 7 with the q you like

```

determines 2,3 generating pairs of $PSL\_2(q)$ up to automorphisms of $PSL$. For $q=71$ I get 30 such pairs and the orders of the product of the generators can be 7,9,12,18,35,36,71. 


