In fact, I want to use the result in GAP to construct the corresponding matrix group. Considering that GAP only support rationalized form in this case, I have to do this kind of treatment. The above data comes from here, and I want to check if this group is isomorphic to U3(3). The complete GAP code is as follows:
gap> G:=Group([[[0, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1/2, -1/2, -1/2, -1/2], [0, 0, 0, 1/2, -1/2, 1/2, -1/2], [0, 0, 0, 1/2, -1/2, -1/2, 1/2], [0, 0, 0, 1/2, 1/2, -1/2, -1/2]],
> [[0, 0, 0, 0, 1, 0, 0], [0, -1/2, -1/2, 0, 0, 1/2, 1/2], [0, 1/2, -1/2, 0, 0, -1/2, 1/2],
> [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, -1/2, 1/2, 0, 0, -1/2, 1/2],
> [0, -1/2, -1/2, 0, 0, -1/2, -1/2]],
> [[0, 0, 0, -1, 0, 0, 0], [0, -1/2, -1/2, 0, 0, -1/2, 1/2], [0, 1/2, -1/2, 0, 0, -1/2, -1/2], [0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 1/2, 1/2, 0, 0, -1/2, 1/2], [0, -1/2, 1/2, 0, 0, -1/2, -1/2]]]);
<matrix group with 3 generators>
gap> gensU33:=[
> [[-1,0,0,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [0,1,0,0,0,1,0],
> [-1,0,0,0,0,0,1],
> [0,0,0,0,1,0,0],
> [0,1,1,0,0,0,0],
> [-1,0,0,1,0,0,0]],
> [[0,-1,0,0,0,-1,0],
> [0,1,1,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [1,0,0,0,0,0,0],
> [0,0,0,0,0,0,-1],
> [0,0,0,-1,0,0,1],
> [0,0,0,0,1,0,1]]];
[ [ [ -1, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 1, 0 ], [ -1, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 1, 1, 0, 0, 0, 0 ], [ -1, 0, 0, 1, 0, 0, 0 ] ], [ [ 0, -1, 0, 0, 0, -1, 0 ], [ 0, 1, 1, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, -1, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 1 ] ] ]
gap> U33:=Group(gensU33);
<matrix group with 2 generators>
gap> IsomorphismGroups(U33,G);
#I Forcing finiteness test
CompositionMapping( [ (1,9)(3,49)(4,5)(6,53)(7,59)(8,10)(13,18)(15,21)(16,57)(17,23)(20,27)(24,58)(25,36)(28,46)(29,41)(31,43)(32,62)(33,44)(34,61)(35,47)(39,55)(40,
50)(45,48)(54,56), (1,40,63,51,2,52,57)(3,46,34,9,10,59,43)(4,58,18,5,6,37,39)(7,47,11,55,23,54,8)(12,14,48,31,49,15,30)(13,17,20,25,27,44,24)(16,41,36,56,50,22,
19)(21,60,38,45,35,26,62)(28,61,29,33,53,42,32) ] -> [ [ [ 0, 0, 0, 1/2, -1/2, 1/2, 1/2 ], [ 0, -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, -1/2, -1/2, 1/2, -1/2 ],
[ 1/2, 0, -1/2, 0, -1/2, -1/2, 0 ], [ -1/2, 0, -1/2, -1/2, 0, 0, 1/2 ], [ 1/2, 0, 1/2, -1/2, 0, 0, 1/2 ], [ 1/2, 0, -1/2, 0, 1/2, 1/2, 0 ] ],
[ [ 0, -1/2, 1/2, 0, 0, -1/2, -1/2 ], [ 1/2, 0, -1/2, 0, 1/2, -1/2, 0 ], [ 1/2, 1/2, 0, 0, -1/2, 0, -1/2 ], [ 1/2, -1/2, 0, 0, -1/2, 0, 1/2 ],
[ 1/2, 0, 1/2, 0, 1/2, 1/2, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ], [ 0, -1/2, -1/2, 0, 0, 1/2, -1/2 ] ] ], <action isomorphism> )
BTW, for other rational numbers, is there a general method to solve this format conversion problem?
Regards,
Zhao