//Standard generators of L3(4) are a and b where a has order 2, b has
//order 4, ab has order 7 and abb has order 5.
//Standard generators of the sextuple cover 6.L3(4) are preimages A and B
//where AB has order 2, B has order 4, AB has order 21, ABB has order 5 and
//ABABABBB has order 5.
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
_LR`AI := [ [ a, b^2*a*b^2*a*b*a*b*a*b^-1*a*b^2*a*b^2 ],
       //L34.2_1 = field x duality - order 2 - not same as in Online ATLAS.
            [ a, (b^-1*a)^3*(b*a)^3*b ],
       //L34.2_2 = field  - order 2 - not same as in Online ATLAS.
            [ a, b*a*(b^-1*a)^4*b^2*a*b*a*b^-1*a*b^-1]  ]
       //L34.2_3 = duality  - order 2q = _1 x _2
                  where a is (_LR`F).1 where b is (_LR`F).2;

//four constituents, permuted by autos
_LR`G :=
/*
Original group: c9Group("6l34p")
Direct induction from degree 2
Schur index: 1
Character: ( 240, -240, 16, -16, -120, -120, 0, 0, 0, 0, 0, 0, 0, 120, 120, 8, 
8, -8, -8, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, -1, -1, 
-1, -1, 0, 0, 0, 0, 1, 1, 1, 1 )
*/

MatrixGroup<240,IntegerRing() |
Matrix(SparseMatrix(240,240,\[
1,3,1,1,4,1,1,1,1,1,2,1,1,9,1,1,10,1,1,13,1,1,
14,1,1,5,1,1,6,1,1,19,1,1,20,1,1,7,1,1,8,1,1,25,1,
1,26,1,1,29,1,1,30,1,1,11,1,1,12,1,1,35,1,1,36,1,1,
37,1,1,38,1,1,15,1,1,16,1,1,27,-1,1,28,-1,1,17,1,1,
18,1,1,45,1,1,46,1,1,49,1,1,50,1,1,21,1,1,22,1,1,
23,1,1,24,1,1,57,1,1,58,1,1,61,1,1,62,1,1,65,1,1,
66,1,1,31,1,1,32,1,1,71,1,1,72,1,1,33,1,1,34,1,1,
75,1,1,76,1,1,79,1,1,80,1,1,83,1,1,84,1,1,39,1,1,
40,1,1,89,1,1,90,1,1,41,1,1,42,1,1,93,1,1,94,1,1,
43,1,1,44,1,1,99,1,1,100,1,2,103,1,104,1,1,103,-1,
1,47,1,1,48,1,1,109,-1,1,110,-1,1,51,1,1,52,1,1,
115,1,1,116,1,1,53,1,1,54,1,1,119,1,1,120,1,1,55,1,
1,56,1,1,125,1,1,126,1,1,129,1,1,130,1,1,59,1,1,
60,1,1,107,1,1,108,1,1,63,1,1,64,1,1,97,1,1,98,1,
1,95,1,1,96,1,1,67,1,1,68,1,1,101,-1,1,102,-1,1,
70,-1,2,69,1,70,1,1,137,-1,1,138,-1,1,91,1,1,92,1,
1,73,-1,1,74,-1,1,145,1,1,146,1,1,149,1,1,150,1,1,
77,1,1,78,1,1,153,-1,1,154,-1,1,81,1,1,82,1,1,157,1,
1,158,1,1,159,1,1,160,1,1,85,1,1,86,1,1,135,1,1,
136,1,1,87,1,1,88,1,1,165,1,1,166,1,1,169,1,1,170,1,
1,127,1,1,128,1,1,105,-1,1,106,-1,1,175,1,1,176,1,
1,177,1,1,178,1,1,182,1,2,181,-1,182,-1,1,111,1,1,
112,1,1,185,1,1,186,1,1,113,1,1,114,1,1,151,-1,1,
152,-1,1,117,-1,1,118,-1,1,189,1,1,190,1,1,121,1,1,
122,1,1,123,1,1,124,1,1,191,1,1,192,1,1,163,-1,1,
164,-1,1,131,1,1,132,1,1,195,1,1,196,1,1,133,1,1,
134,1,2,199,-1,200,-1,1,199,1,2,201,1,202,1,1,201,-1,
1,139,1,1,140,1,1,141,1,1,142,1,1,207,1,1,208,1,2,
143,-1,144,-1,1,143,1,1,209,1,1,210,1,1,147,1,1,148,1,
1,187,-1,1,188,-1,1,155,1,1,156,1,1,161,1,1,162,1,
1,213,1,1,214,1,1,167,1,1,168,1,1,205,1,1,206,1,1,
172,1,2,171,-1,172,-1,1,174,-1,2,173,1,174,1,1,203,-1,
1,204,-1,1,197,1,1,198,1,1,179,1,1,180,1,1,183,1,1,
184,1,2,221,1,222,1,1,221,-1,1,193,1,1,194,1,1,223,1,
1,224,1,1,232,1,2,231,-1,232,-1,1,219,-1,1,220,-1,1,
212,-1,2,211,1,212,1,1,215,1,1,216,1,2,227,-1,228,-1,
1,227,1,1,226,1,2,225,-1,226,-1,1,233,-1,1,234,-1,2,
217,-1,218,-1,1,217,1,1,229,-1,1,230,-1,1,235,-1,1,
236,-1,1,239,1,1,240,1,1,237,1,1,238,1
])),Matrix(SparseMatrix(240,240,\[
1,5,1,1,6,1,1,7,1,1,8,1,1,11,1,1,12,1,1,15,1,1,
16,1,1,17,1,1,18,1,1,21,1,1,22,1,1,23,1,1,24,1,1,
27,1,1,28,1,1,31,1,1,32,1,1,33,1,1,34,1,1,1,1,1,2,1,
1,39,1,1,40,1,1,41,1,1,42,1,1,3,1,1,4,1,2,43,1,44,1,
1,43,-1,1,47,1,1,48,1,1,51,1,1,52,1,1,53,1,1,54,1,
1,55,1,1,56,1,1,59,1,1,60,1,1,63,1,1,64,1,1,67,1,
1,68,1,1,69,1,1,70,1,1,9,1,1,10,1,1,73,1,1,74,1,1,
77,1,1,78,1,1,81,1,1,82,1,1,85,1,1,86,1,1,88,1,2,
87,-1,88,-1,1,13,1,1,14,1,1,91,1,1,92,1,1,95,1,1,
96,1,1,97,1,1,98,1,1,102,-1,2,101,1,102,1,1,105,1,
1,106,1,1,107,1,1,108,1,2,111,-1,112,-1,1,111,1,1,
113,1,1,114,1,1,19,1,1,20,1,1,117,1,1,118,1,1,121,1,
1,122,1,2,123,-1,124,-1,1,123,1,1,127,1,1,128,1,2,
57,1,58,1,1,57,-1,1,131,1,1,132,1,1,61,-1,1,62,-1,1,
133,1,1,134,1,1,25,1,1,26,1,1,129,1,1,130,1,1,136,-1,
2,135,1,136,1,1,29,1,1,30,1,1,110,-1,2,109,1,110,1,
1,140,-1,2,139,1,140,1,1,141,1,1,142,1,2,143,1,144,1,
1,143,-1,1,147,1,1,148,1,1,93,1,1,94,1,1,151,-1,1,
152,-1,1,79,-1,1,80,-1,1,155,-1,1,156,-1,1,35,1,1,
36,1,2,115,-1,116,-1,1,115,1,1,162,1,2,161,-1,162,-1,
1,37,1,1,38,1,1,163,1,1,164,1,1,167,1,1,168,1,1,
75,1,1,76,1,1,171,1,1,172,1,1,174,-1,2,173,1,174,1,
2,45,1,46,1,1,45,-1,1,179,1,1,180,1,1,183,1,1,184,1,
2,99,1,100,1,1,99,-1,1,50,1,2,49,-1,50,-1,1,165,1,
1,166,1,2,83,1,84,1,1,83,-1,2,187,-1,188,-1,1,187,1,
1,137,1,1,138,1,1,153,1,1,154,1,1,157,1,1,158,1,2,
193,-1,194,-1,1,193,1,1,65,1,1,66,1,1,149,-1,1,150,-1,
1,197,1,1,198,1,1,175,1,1,176,1,1,145,1,1,146,1,2,
119,-1,120,-1,1,119,1,2,203,1,204,1,1,203,-1,2,205,-1,
206,-1,1,205,1,1,71,1,1,72,1,2,207,-1,208,-1,1,207,1,
1,103,1,1,104,1,2,211,-1,212,-1,1,211,1,1,160,1,2,
159,-1,160,-1,2,201,-1,202,-1,1,201,1,2,169,1,170,1,
1,169,-1,1,216,1,2,215,-1,216,-1,1,217,1,1,218,1,
1,89,1,1,90,1,2,219,-1,220,-1,1,219,1,1,221,-1,1,
222,-1,2,191,-1,192,-1,1,191,1,1,199,1,1,200,1,1,
223,-1,1,224,-1,2,225,-1,226,-1,1,225,1,1,227,1,1,
228,1,1,229,1,1,230,1,2,125,-1,126,-1,1,125,1,1,213,1,
1,214,1,2,177,-1,178,-1,1,177,1,1,233,1,1,234,1,2,
235,-1,236,-1,1,235,1,1,238,1,2,237,-1,238,-1,1,232,1,
2,231,-1,232,-1,1,195,1,1,196,1,1,185,1,1,186,1,1,
190,-1,2,189,1,190,1,2,181,1,182,1,1,181,-1,2,239,-1,
240,-1,1,239,1,1,210,1,2,209,-1,210,-1
]))>;

return _LR;
