[GAP Forum] extending groups to isomorphics acting on a multiple of domain

[Rudolf Zlabinger]

Dear GAP Forum,

I try to extend the number of points acted on by permutation groups to multiples of the original domain.
For example i tried to find a subgroup of S60 isomorphic to A5, thus acting on 60 points.
As the samples attached show, I did it by brute force, extending the generators by acting on all residue classes mod the multiplier.

This method yields subgroups of the desired target symmetric groups, but only in one subgroup conjugacy class. The conjugacy class(es) I am looking for, are those, whose subgroups are transitive, as needed, for example, to be true rotation groups acting on the vertices of a solid.

This was possible in the case of A5 extending to S12, but not to S30 or even S60, as those targets are to big for executing, for example IsomorphicSubgroups.

In the case of finite rotation groups there are other methods using presentations in Linear Algebra too. I used blueprints of solids to get the right generator permutations for S12 and S60 for the desired rotation groups. 

1. Is there a simpler method for extending the domain acted on to a multiple in general?

2. Is there a feasable method for big Symmetric Groups as S60 to find conjugacy classes of isomorphic subgroups (for example for A5) beeing transitive?

Thank you and best regards, Rudolf Zlabinger
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# Two examples of extending the points acted on to a set of a multiple Size
# The method is to assign the original points the mods 1 of the multiplier
# and extending the generators by cycles consisting of the rest of the mods
# The result yields a isomorphic subgroup of the symmetric group (nr of  points)

# It may be not the desired conjugacy class, thus a finite rotation group may not be
# preserved. In the case of finite rotation groups one has to look for a 
# conjugacy class of transitive groups, i.e a group whose orbit on a point is the full set 
# of points 



# First we extend a group acting on 12 points to a group acting on 60 points




#   Thats the group covering a "canonical" label set of icosahedron
#   gencgroup:=GeneratorsOfGroup(cgroup);
#   [ (1,2,3)(4,6,7)(5,11,8)(9,10,12), (1,2)(3,6)(4,11)(5,7)(8,10)(9,12) ]
#   cgroup itself is not transitive, and therefore itself not a true rotation group

#   we calculate the generators as acting on numbers mod(5), where the mods 1 are the 
#   original
#   points of the icosahedron
 
#    list0010:=[1,2,3,4,6,7,5,11,8,9,10,12];
#    list0020:=[1,2,3,6,4,11,5,7,8,10,9,12];
#    list0030:=[1,2,3,4,5];
#    result0010:=List( list0030,c->List( list0010,d->( (d-1)*5+c) ) );
#    result0020:=List( list0030,c->List( list0020,d->( (d-1)*5+c) ) );

	
#   result0010;
#   [ [ 1, 6, 11, 16, 26, 31, 21, 51, 36, 41, 46, 56 ], 
#   [ 2, 7, 12, 17, 27, 32, 22, 52, 37, 42, 47, 57 ], 
#   [ 3, 8, 13, 18, 28, 33, 23, 53, 38, 43, 48, 58 ], 
#   [ 4, 9, 14, 19, 29, 34, 24, 54, 39, 44, 49, 59 ], 
#   [ 5, 10, 15, 20, 30, 35, 25, 55, 40, 45, 50, 60 ] ]
#   result0020;
#   [ [ 1, 6, 11, 26, 16, 51, 21, 31, 36, 46, 41, 56 ], 
#   [ 2, 7, 12, 27, 17, 52, 22, 32, 37, 47, 42, 57 ], 
#   [ 3, 8, 13, 28, 18, 53, 23, 33, 38, 48, 43, 58 ], 
#   [ 4, 9, 14, 29, 19, 54, 24, 34, 39, 49, 44, 59 ], 
#   [ 5, 10, 15, 30, 20, 55, 25, 35, 40, 50, 45, 60 ] ]

#   So we get as generators of our icosahedron acting on 60 points

    gen_0010:=  (1, 6, 11)  (16, 26, 31) (21, 51, 36) (41, 46, 56)  
                (2, 7, 12)  (17, 27, 32) (22, 52, 37) (42, 47, 57)  
                (3, 8, 13)  (18, 28, 33) (23, 53, 38) (43, 48, 58)  
                (4, 9, 14)  (19, 29, 34) (24, 54, 39) (44, 49, 59)  
                (5, 10, 15) (20, 30, 35) (25, 55, 40) (45, 50, 60);

    gen_0020:=  (1, 6)  (11, 26) (16, 51) (21, 31) (36, 46) (41, 56)  
                (2, 7)  (12, 27) (17, 52) (22, 32) (37, 47) (42, 57)  
                (3, 8)  (13, 28) (18, 53) (23, 33) (38, 48) (43, 58)  
                (4, 9)  (14, 29) (19, 54) (24, 34) (39, 49) (44, 59)  
                (5, 10) (15, 30) (20, 55) (25, 35) (40, 50) (45, 60);

    group0010:=Group(gen_0010,gen_0020);

#    StructureDescription(group0010);
#    "A5"                             It is isomorphic indeed
#    NrMovedPoints(group0010);
#    60                               and acting on 60 points as desired


# The next sample extends A5 to a isomorphic subgroup of S30


# Generators of A5     (1,3,2), (2,4,3), (2,3)(4,5)

gens0050:= ( 1,13, 7)( 2,14, 8)( 3,15, 9)( 4,16,10)( 5,17,11)( 6,18,12);
gens0051:= ( 7,19,13)( 8,20,14)( 9,21,15)(10,22,16)(11,23,17)(12,24,18);
gens0052:= ( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)
            (19,25)(20,26)(21,27)(22,28)(23,29)(24,30);

group0050:=Group(gens0050,gens0051,gens0052);