Equivalence classes of (2,3)-pairs in PSL(2,q) This may not be a research-level question, which is why I submitted it to math.stackexchange first, but so far the question there has barely been viewed, let alone answered. Apologies if submitting it here as well is poor etiquette.

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.
 A: 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. 
A: While your observation is true also for ${\rm PSL}(2,7)$, ${\rm PSL}(2,8)$
and ${\rm PSL}(2,9)$, it is false for ${\rm PSL}(2,11)$. --
In ${\rm PSL}(2,11)$ there are two orbits of $(2,3)$-pairs whose product
has order $5$, as you can check with GAP as follows:
gap> G := PSL(2,11);;
gap> A := AutomorphismGroup(G);;
gap> pairs := Cartesian(Filtered(AsList(G),g->Order(g)=2),
>                       Filtered(AsList(G),g->Order(g)=3));;
gap> orbits := Orbits(A,pairs,OnTuples);;
gap> List(orbits,orbit->Order(Product(orbit[1])));
[ 5, 5, 11, 3, 2, 6 ]

A: The paper "Arc-transitive cubic cayley graphs on PSL(2,p)" by Du and Wang seems to give a definitive answer to my question for the case where q is prime.
