//Standard generators of L2(19) are a and b where a has order 2, b has order 3 
//and ab has order 19.
//Standard generators of the double cover 2.L2(19) = SL2(19) are preimages A 
//and B where B has order 3 and AB has order 19.
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
_LR`AI := [ [a^-1, b^-1] ]
             where a is (_LR`F).1 where b is (_LR`F).2;
//two constituents interchanged by _LR`AI[1][1]
_LR`G :=
MatrixGroup<9, ext<K|Polynomial(K, [5, 1, 1])> where K is RationalField() |
[[-4,38],[-63,-11],[
-86,-6],[-78,-17],[
-90,-18],[91,28],[
-52,-14],[82,4],[
-4,3],[1,-14],[23,4],[
31,2],[29,6],[
34,6],[-33,-10],[19,5],[
-30,-1],[2,-1],[
0,0],[1,0],[1,0],[
0,0],[0,0],[
-1,0],[0,0],[1,0],[
0,0],[-23,4],[
-9,-9],[-17,-11],[-8,-11],[
-11,-12],[6,14],[
-4,-8],[16,10],[-3,0],[
-22,-8],[12,-5],[
11,-9],[16,-6],[18,-7],[
-24,5],[13,-3],[
-9,9],[-2,-1],[10,-15],[
26,7],[37,6],[
31,10],[36,11],[-34,-15],[
20,8],[-35,-5],[
2,-1],[-22,-5],[6,-6],[
4,-9],[10,-7],[
10,-8],[-16,7],[8,-4],[
-3,9],[-3,-1],[
9,-1],[2,3],[6,4],[
2,4],[2,5],[0,
-5],[1,3],[-6,-4],[
0,0],[17,-10],[
18,8],[28,9],[21,11],[
24,12],[-21,-15],[
11,8],[-27,-8],[
3,-1]],
[[80,24],[-31,20],[
-25,34],[-48,23],[
-55,27],[71,-22],[
-39,13],[24,-33],[
7,3],[22,0],[
2,8],[7,10],[0,9],[
2,10],[4,-11],[
-2,6],[-7,-9],[3,0],[
-9,1],[-2,-3],[
-6,-4],[-2,-4],[-2,-5],[
0,5],[-1,-3],[
6,4],[0,0],[-24,13],[
-23,-11],[-37,-12],[
-27,-15],[-32,-17],[
26,20],[-16,-11],[
37,11],[-3,1],[
0,0],[0,0],[-1,0],[
0,0],[0,0],[0,
0],[0,0],[0,0],[
0,0],[-4,-9],[
14,1],[18,-1],[18,2],[
22,2],[-22,-4],[
12,2],[-18,1],[1,-1],[
-27,10],[-20,-12],[
-32,-14],[-21,-15],[
-25,-16],[20,20],[
-11,-11],[29,12],[
-4,1],[-10,-8],[
12,-2],[15,-4],[16,-1],[
17,-1],[-21,0],[
11,0],[-14,3],[-1,-1],[
-25,-5],[7,-7],[
3,-11],[10,-8],[10,-10],[
-17,9],[9,-5],[
-2,10],[-2,-1]]>;
return _LR;
