primitivity of permutation group

Question

If we are to check that extended triangle group $(p,q,r)=<x,y,t: x^p=y^q =t^2=(xy)^r=(xt)^2=(yt)^2=1>$ is primitive, how we can check it in GAP if we have permutation representation of x,y and t already found in GAP for some p, q, r of index subgroup of extended triangle group (p,q,r)?

Secondly, I am confused, what does it mean "G contains a prime cycle v" in Jordans theorem? Is this cycle structure of xy or xyt or $(x^{-1}y)$ in group G? In B. Everitts research papers of "Alternating qoutiets of Fuchsian groups" he consider cycle structure of xy or xyt? I am confused in making sense of the cycle structure for what we are looking for to check the cycle of length $p$ in group G. I am sorry this might be stupid question regarding prime cycle, however, I could not get the concept behind it. I shall be very thankful if someone could assist me in this aspect.

I tried something like this, that I am sure is wrong something with it

PrimitiveGroup:=function(perms)

local n,m,x,y,t,g,f;

x:=perms[1];

t:=perms[2];

y:=perms[3];

f:=FreeGroup(3);

g:=f/[x^3,y^7,t^2,(xt)^2,(yt)^2,(xy)^9];

n:=Order(g);

m:=IsPrimitive(g);

Print("Is this group g primitive",m,"\n");

end;

here perms is the permutation represtation that I already got in some program but in the form of as an example like this "[(2,3,4)(5,6,7),(1,2)(3,4),(1,2,3,4,5,6,7)]".

I am not sure how could I get the form of fp group from here. However, this is what I have in my mind.

Answer 1

Just summarising the outcome of the discussion in the comments - this is how to check in GAP that $G$ is a primitive permutation group, and in fact it is a natural alternating group on the points $ \{ 1, ..., 7 \}$:

gap> G:=Group((2,3,4)(5,6,7),(1,2)(3,4),(1,2,3,4,5,6,7));
Group([ (2,3,4)(5,6,7), (1,2)(3,4), (1,2,3,4,5,6,7) ])
gap> IsPrimitive(G,MovedPoints(G));
true
gap> IsAlternatingGroup(G);
true
gap> IsNaturalAlternatingGroup(G);
true