[GAP Forum] subgroups of index 4 up to conjugacy

Levie Bicua lee_bkua at yahoo.com
Thu Nov 27 10:27:39 GMT 2008

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.



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