//Standard generators of U3(5) are a and b where a has order 3,
//b is in class 5A and ab has order 7.
//Standard generators of 3.U3(5) are preimages A and B where B has order 5 and
//AB has order 7.
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;

_LR`AI := [[a,a*b^-1*a*b^2*a*b*a*b*a*b^2*a*b^-1*a*b],
       //L34.2_1 = field - order 2.
           [a,(b^-1*a^-1*b^-1*a^-1*b^-1*a*b*a*b)^2]]
       //L34.3 = diagonal - order 3
                  where a is (_LR`F).1 where b is (_LR`F).2;

//one constituent, fixed by _LR`AI[1], _LR`AI[2]
_LR`G := sub<GL(21,Integers()) |
\[ 1,0,0,0,-1,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,1,-1,
0,1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,
0,1,1,1,0,1,0,-1,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,1,
-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,
-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,
0,0,0,-1,0,0,0,-1,0,-1,0,0,-1,-1,0,0,-1,-1,1,0,0,0,0,
0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,-1,1,0,0,0,0,0,-1,0,
0,0,1,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,1,0,
-1,0,0,0,0,0,0,0,0,0,1,1,0,0,-1,1,-1,0,0,0,0,0,0,0,
0,0,0,0,1,-1,0,1,1,0,0,-1,1,-1,0,0,0,-1,0,0,0,0,-1,0,
-1,0,0,-1,1,0,0,0,1,0,0,0,0,1,0,0,-1,0,0,-1,-1,0,0,0,
0,0,0,1,0,0,0,0,0,-1,2,1,-1,1,0,0,0,0,1,0,1,0,0,1,1,
0,-1,0,0,1,0,-1,-1,0,-1,0,0,-1,0,-1,0,0,0,0,-1,0,0,1,
0,0,0,-1,1,0,-1,0,-1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,
-1,1,0,-1,0,-1,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,-1,1,0,1,1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,
0,0,0,-1,0,0 ],
\[ 0,0,0,0,0,0,-1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,
0,0,0,-1,1,1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,
-1,-1,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,-1,0,1,-1,-1,0,-1,-1,
-1,1,0,0,0,0,0,0,-1,0,-1,0,0,1,0,0,1,0,1,0,0,1,-1,0,
-1,0,0,0,0,0,0,-1,0,0,0,0,-1,1,1,0,0,0,0,-1,0,0,0,0,
0,0,0,0,-1,0,0,1,1,0,1,1,1,-1,0,0,-1,0,0,-1,0,0,0,0,
0,0,0,-1,-1,-1,-1,0,0,-1,1,1,0,0,0,0,0,0,0,0,0,0,0,
-1,0,0,0,0,-1,-1,-1,1,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,
0,0,0,0,0,0,-1,0,0,-1,-1,0,-1,1,0,0,0,0,0,1,1,0,0,0,
0,0,-1,1,0,1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,-1,-1,0,0,-1,-1,
0,0,0,0,1,0,0,0,0,0,1,0,0,0,-1,1,1,-1,1,1,1,0,0,0,1,
-1,0,0,0,-1,0,-1,0,0,1,1,-1,0,0,-1,-1,-1,0,0,0,0,0,0,
0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,1,0,0,-1,1,0,0,0,0,0,
-1,0,0,0,0,0,0,0,0,1,0,1,-1,0,0,0,0,0,0,0,0,-1,0,0,
-1,0,-1,0,0,-1,1,0,1,0,0,0,0,0,0,0,0,0,-1,-1,-1,-1,0,
-1,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,1,0,0,1,1,
1,0,0,0,0,0,0,0,0,0,-1,0 ] >;

return _LR;
