Dear GAP forum members,
Thank you for accepting me in this forum. I am working on determining all
the index 5 subgroups of the triangle group H=*pqr=<P,Q,R> where P,Q, and R are
the sides of the triangle. To do this, I will construct all 4-colorings of the
Tiling T of the plane (the triangle with sides P,Q,R is the fundamental region
of T) where all elements of H effect permutations of the 4 colors {1,2,3,4}.
For such colorings I defined a homomorphism pi:H->S4, D4 and V (the symmetric,
dihedral, and Klein-4 group respectively) which are transitive subgroups of S4.
Suppose pi:H->S4, since H=<P,Q,R>, pi is completely determined when pi(P),
pi(Q) and pi(R) are specified. Since H is to permute the colors in the
resulting coloring, each of P,Q,R either fixes or interchanges any two colors;
i.e., each of P,Q,R can be mapped to any of the 2-cycle or products of 2-cycle
element of s4.
If I want only the index 4 subgroups of H when pi(H)=S4, the first step
would be to list all the generators of the symmetry group S4 with each
generator consisting only of 3 elements from the 9 2-cyles and products of two
cycles of S4 (e.g.{(12),(13),(24)}, {(13),(14)(23),(12)},…). I know that it's
easier to do this using GAP but I just don't know how. Can you give me a
working program or code for this? Thanks.
Now, suppose {pi(P),pi(Q),pi(R)} is a permutation assignment to P,Q,R that
gives rise to an index 4 subgroup K in H. The entries corresponding to
(1234){pi(P),pi(Q),pi(R)}(1234)^-1,
(1432){pi(P),pi(Q),pi(R)}(1432)^-1, and
(13)(24){pi(P),pi(Q),pi(R)}((12)(34))^-1 will respectively, yield h1Kh1^-1,
h2Kh2^-1, h3Kh3^-1 (for some h1,h2,h3 in H), conjugate subgroups of K in H. My
next question is, how do I use GAP to obtain the distinct subgroups of index 4
in H up to conjugacy when pi(H)=S4?
Many thanks.
Levi
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