[GAP Forum] Ikosaeder group

[Rudolf Zlabinger]

I dealt with the finite rotation group for the ikosaeder.

My task was to find a permutation group acting on the 12 vertices of
ikosaeder recognizing a special numbering of the vertices as such: look at
the ikosaeder from a top vertex, numbered as 1, then on the 2 x 5 vertices
forming 2 somehow  parallel cycles numbered by 2,3,4,5,6 and 7,8,9,10,11,
followed by the bottom vertex numbered by 12. Thus the permutation group
looked for should contain the rotation around the axis 1,12 executed by the
permutation
(2,3,4,5,6)(7,8,9,10,11).

Unfortunately the representative isomorphism Groups of symmetric group 12
itself did not contain the desired rotation, so I had to look for a
conjugate group of them containing it. I for the first glance found no
direct method for doing that, so I used the following sequence:



s12:=SymmetricGroup(12);

     normact:=(2,3,4,5,6)(7,8,9,10,11);;
     # the desired rotation

     ccnormact:=ConjugacyClass(s12,normact);
     # the conjugacyclass containing the rotation

     Read(Ikosaeder);
     # ikosaeder is introduced as group of perm matrices on 12 points
     # representing the vertices of ikosaeder numbered as desired

     isos12:=IsomorphicSubgroups(s12,ikosaeder);
     # to mirror the matrices to permgroups acting on 12 points

     iisos12:=List(isos12,c->Image(c));
     # to get groups; none of the representative groups contains normact
<=======

     enumiisos12:=List(iisos12,c->Enumerator(c));
     # to prepare for efficiency

     intersects:=List(enumiisos12,c->Intersection(c,ccnormact));
     # to get intersects of groups to conjugacy class containing the desired
rotation

     grpact:=(2,3,6,7,4)(5,9,12,10,8);
     # one permutation of the intersect of the 7th group

     repop70:=RepresentativeAction(s12,grpact,normact);
     # to get the desired conjugator

     cgroup70:=iisos12[7]^repop70;
     # to get the desired conjugated group

#    normact in cgroup70;  the desired rotation is finally in the resulting
group
#    true

My question to you is: Is there a more direct method for doing the above?

Thank you, friendly greetings from Vienna, Rudolf Zlabinger