_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
//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 cover 12a.L3(4) are preimages A and B
//where A has order 2, B has order 4, AB has order 21 and ABB has order 5.

_LR`AI :=
    [[a,a*b^-1*a*b*a*b^-1*a*b*a*b*a*b^-1*a*b*a*b*a*b^-1*a*b*a*b^-1*(a*b)^7],
       //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^-1]  ]
       //L34.2_3 = duality  - order 2
                  where a is (_LR`F).1 where b is (_LR`F).2;

//four constituents, not fixed by any automorphism.
_LR`G :=
/*
Original group: c9Group("12al34p")
Direct induction from degree 2
Schur index: 1
Character: ( 480, -480, 0, -240, -240, 0, 0, 0, 0, 0, 0, 0, 0, 240, 240, 0, 0, 
0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, 0, 0, 
0, 0, 0, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
*/

MatrixGroup<480,IntegerRing() |
Matrix(SparseMatrix(480,480,\[
1,1,-1,1,2,-1,1,5,1,1,6,1,1,3,1,1,4,1,1,23,1,1,
24,1,1,31,1,1,32,1,1,41,1,1,42,1,1,49,1,1,50,1,
1,55,1,1,56,1,1,63,1,1,64,1,1,71,1,1,72,1,1,79,1,
1,80,1,1,7,1,1,8,1,1,53,1,1,54,1,1,65,1,1,66,1,1,
101,1,1,102,1,1,9,1,1,10,1,1,113,1,1,114,1,1,45,1,1,
46,1,1,125,1,1,126,1,1,85,1,1,86,1,1,11,1,1,12,1,1,
143,1,1,144,1,1,35,1,1,36,1,1,149,1,1,150,1,1,13,1,
1,14,1,1,51,1,1,52,1,1,25,1,1,26,1,1,15,1,1,16,1,
1,169,1,1,170,1,2,111,-1,112,-1,1,111,1,1,129,1,1,
130,1,1,17,1,1,18,1,1,27,1,1,28,1,1,109,1,1,110,1,
1,189,1,1,190,1,1,19,1,1,20,1,1,197,1,1,198,1,1,
139,1,1,140,1,1,131,1,1,132,1,1,21,1,1,22,1,1,81,-1,
1,82,-1,1,147,1,1,148,1,1,39,1,1,40,1,1,225,1,1,
226,1,1,96,1,2,95,-1,96,-1,1,233,1,1,234,1,2,193,1,
194,1,1,193,-1,2,89,-1,90,-1,1,89,1,1,242,-1,2,241,1,
242,1,1,99,1,1,100,1,1,29,1,1,30,1,1,249,1,1,250,1,1,
185,1,1,186,1,2,259,-1,260,-1,1,259,1,1,67,1,1,68,1,
1,60,1,2,59,-1,60,-1,1,33,1,1,34,1,1,211,1,1,212,1,
1,281,1,1,282,1,1,289,1,1,290,1,1,295,1,1,296,1,2,
303,1,304,1,1,303,-1,1,37,1,1,38,1,1,135,1,1,136,1,1,
61,1,1,62,1,1,77,1,1,78,1,1,191,1,1,192,1,1,127,1,
1,128,1,1,325,1,1,326,1,1,75,1,1,76,1,1,313,1,1,
314,1,1,43,1,1,44,1,1,287,1,1,288,1,1,83,1,1,84,1,1,
47,1,1,48,1,1,151,1,1,152,1,1,323,1,1,324,1,1,335,1,
1,336,1,1,341,1,1,342,1,1,159,-1,1,160,-1,1,247,-1,
1,248,-1,1,275,1,1,276,1,1,224,1,2,223,-1,224,-1,1,
181,1,1,182,1,1,57,1,1,58,1,1,353,1,1,354,1,1,173,-1,
1,174,-1,1,359,1,1,360,1,1,363,1,1,364,1,1,365,1,
1,366,1,1,167,1,1,168,1,1,251,1,1,252,1,1,105,1,1,
106,1,1,227,1,1,228,1,1,69,1,1,70,1,1,133,1,1,134,1,
1,94,-1,2,93,1,94,1,1,229,1,1,230,1,1,73,1,1,74,1,1,
343,1,1,344,1,1,318,-1,2,317,1,318,1,1,385,1,1,386,1,
2,387,1,388,1,1,387,-1,1,389,-1,1,390,-1,1,209,-1,
1,210,-1,1,115,1,1,116,1,1,399,1,1,400,1,1,215,1,
1,216,1,1,217,1,1,218,1,1,405,1,1,406,1,1,409,1,1,
410,1,2,165,-1,166,-1,1,165,1,1,87,1,1,88,1,1,187,1,1,
188,1,1,195,1,1,196,1,1,375,1,1,376,1,1,91,1,1,92,1,
1,431,1,1,432,1,1,263,1,1,264,1,1,333,1,1,334,1,2,
97,1,98,1,1,97,-1,1,268,-1,2,267,1,268,1,1,437,1,1,
438,1,1,161,-1,1,162,-1,1,103,1,1,104,1,1,183,1,1,
184,1,2,443,1,444,1,1,443,-1,1,309,1,1,310,1,1,257,1,
1,258,1,1,108,1,2,107,-1,108,-1,1,369,1,1,370,1,1,
237,1,1,238,1,1,315,1,1,316,1,2,243,1,244,1,1,243,-1,
2,355,-1,356,-1,1,355,1,1,455,1,1,456,1,1,419,1,1,
420,1,1,163,1,1,164,1,1,411,1,1,412,1,1,380,-1,2,
379,1,380,1,1,117,1,1,118,1,1,294,1,2,293,-1,294,-1,
1,459,1,1,460,1,1,145,1,1,146,1,1,119,1,1,120,1,1,
368,-1,2,367,1,368,1,2,283,-1,284,-1,1,283,1,1,121,1,1,
122,1,1,351,1,1,352,1,1,439,1,1,440,1,2,337,1,338,1,1,
337,-1,1,124,-1,2,123,1,124,1,1,446,1,2,445,-1,446,-1,
1,430,-1,2,429,1,430,1,1,255,1,1,256,1,1,311,1,1,
312,1,1,141,1,1,142,1,1,265,1,1,266,1,2,201,1,202,1,
1,201,-1,1,435,1,1,436,1,1,415,1,1,416,1,1,153,1,1,
154,1,1,137,1,1,138,1,2,461,1,462,1,1,461,-1,1,329,-1,
1,330,-1,1,339,1,1,340,1,1,239,1,1,240,1,1,155,1,1,
156,1,1,302,-1,2,301,1,302,1,1,331,1,1,332,1,1,157,1,
1,158,1,1,199,1,1,200,1,1,403,-1,1,404,-1,1,471,-1,
1,472,-1,2,407,1,408,1,1,407,-1,1,297,1,1,298,1,1,
171,1,1,172,1,1,270,1,2,269,-1,270,-1,1,395,1,1,396,1,
1,175,1,1,176,1,1,427,1,1,428,1,1,177,1,1,178,1,1,
179,1,1,180,1,2,291,1,292,1,1,291,-1,1,261,1,1,262,1,
1,383,1,1,384,1,1,417,1,1,418,1,1,231,1,1,232,1,1,
458,1,2,457,-1,458,-1,2,279,1,280,1,1,279,-1,1,451,-1,
1,452,-1,1,371,1,1,372,1,1,203,1,1,204,1,1,206,-1,2,
205,1,206,1,1,207,-1,1,208,-1,2,401,1,402,1,1,401,-1,
1,393,-1,1,394,-1,1,357,1,1,358,1,1,467,1,1,468,1,
1,213,1,1,214,1,1,392,-1,2,391,1,392,1,1,345,-1,1,
346,-1,1,219,1,1,220,1,1,350,-1,2,349,1,350,1,1,221,1,
1,222,1,1,277,1,1,278,1,1,475,1,1,476,1,1,321,1,1,
322,1,1,373,1,1,374,1,1,273,1,1,274,1,1,469,1,1,470,1,
1,463,1,1,464,1,1,447,1,1,448,1,1,361,1,1,362,1,2,
307,1,308,1,1,307,-1,1,235,1,1,236,1,1,477,1,1,478,1,
1,319,1,1,320,1,1,245,1,1,246,1,1,299,1,1,300,1,1,
441,1,1,442,1,1,254,-1,2,253,1,254,1,2,305,-1,306,-1,
1,305,1,1,425,1,1,426,1,1,466,1,2,465,-1,466,-1,1,
381,-1,1,382,-1,1,473,1,1,474,1,1,271,1,1,272,1,2,
377,-1,378,-1,1,377,1,1,285,1,1,286,1,1,328,-1,2,
327,1,328,1,1,423,1,1,424,1,2,449,-1,450,-1,1,449,1,
1,397,1,1,398,1,1,421,1,1,422,1,1,347,-1,1,348,-1,
1,453,1,1,454,1,1,413,1,1,414,1,1,433,1,1,434,1,1,
479,-1,1,480,-1
])),Matrix(SparseMatrix(480,480,\[
1,3,1,1,4,1,1,7,1,1,8,1,1,15,1,1,16,1,1,25,1,1,
26,1,1,33,1,1,34,1,1,21,1,1,22,1,1,51,1,1,52,1,1,
57,1,1,58,1,1,65,1,1,66,1,1,73,1,1,74,1,1,81,1,1,
82,1,1,87,1,1,88,1,1,1,1,1,2,1,1,19,1,1,20,1,1,
103,-1,1,104,-1,1,107,1,1,108,1,1,115,1,1,116,1,2,
123,1,124,1,1,123,-1,2,127,1,128,1,1,127,-1,1,133,1,
1,134,1,2,137,-1,138,-1,1,137,1,2,143,-1,144,-1,1,
143,1,2,145,-1,146,-1,1,145,1,1,151,1,1,152,1,1,155,1,
1,156,1,1,157,1,1,158,1,1,163,1,1,164,1,1,125,1,
1,126,1,1,171,1,1,172,1,1,178,1,2,177,-1,178,-1,2,
179,-1,180,-1,1,179,1,1,182,1,2,181,-1,182,-1,1,183,1,
1,184,1,1,187,-1,1,188,-1,1,191,-1,1,192,-1,1,195,1,1,
196,1,1,199,1,1,200,1,1,207,1,1,208,1,2,209,1,210,1,
1,209,-1,1,211,1,1,212,1,1,213,1,1,214,1,1,221,1,
1,222,1,1,223,1,1,224,1,1,227,1,1,228,1,1,231,1,
1,232,1,1,235,1,1,236,1,1,237,1,1,238,1,1,111,1,
1,112,1,1,243,1,1,244,1,1,245,1,1,246,1,1,61,-1,1,
62,-1,1,29,1,1,30,1,1,255,-1,1,256,-1,1,31,1,1,32,1,
1,267,1,1,268,1,1,265,1,1,266,1,1,269,1,1,270,1,1,
275,1,1,276,1,1,202,-1,2,201,1,202,1,1,131,1,1,132,1,
1,297,1,1,298,1,1,217,1,1,218,1,1,311,1,1,312,1,1,
315,1,1,316,1,1,109,1,1,110,1,1,44,1,2,43,-1,44,-1,
1,317,1,1,318,1,1,323,1,1,324,1,1,193,1,1,194,1,1,
328,-1,2,327,1,328,1,1,329,1,1,330,1,1,119,1,1,120,1,
1,260,1,2,259,-1,260,-1,1,215,1,1,216,1,1,91,1,1,
92,1,2,331,-1,332,-1,1,331,1,1,53,1,1,54,1,1,135,1,
1,136,1,1,319,1,1,320,1,1,289,1,1,290,1,1,36,-1,2,
35,1,36,1,1,343,1,1,344,1,1,349,1,1,350,1,1,229,1,1,
230,1,1,351,1,1,352,1,1,5,1,1,6,1,2,355,-1,356,-1,1,
355,1,2,361,-1,362,-1,1,361,1,1,314,-1,2,313,1,314,1,
1,283,-1,1,284,-1,2,321,-1,322,-1,1,321,1,2,369,-1,
370,-1,1,369,1,1,371,1,1,372,1,1,67,1,1,68,1,2,
241,-1,242,-1,1,241,1,1,69,-1,1,70,-1,2,251,-1,252,-1,
1,251,1,1,169,1,1,170,1,1,114,1,2,113,-1,114,-1,1,
27,1,1,28,1,1,338,-1,2,337,1,338,1,1,97,1,1,98,1,
1,167,1,1,168,1,1,391,1,1,392,1,1,83,1,1,84,1,1,
397,1,1,398,1,1,11,1,1,12,1,1,403,1,1,404,1,1,161,1,
1,162,1,1,45,1,1,46,1,1,78,-1,2,77,1,78,1,1,381,1,
1,382,1,1,359,1,1,360,1,1,419,1,1,420,1,1,423,1,1,
424,1,2,233,1,234,1,1,233,-1,1,263,1,1,264,1,1,277,1,
1,278,1,1,93,-1,1,94,-1,1,121,1,1,122,1,1,436,-1,2,
435,1,436,1,1,293,1,1,294,1,1,429,1,1,430,1,2,357,1,
358,1,1,357,-1,1,225,1,1,226,1,2,41,-1,42,-1,1,41,1,
1,445,1,1,446,1,1,307,1,1,308,1,1,79,1,1,80,1,1,
219,1,1,220,1,1,291,1,1,292,1,1,90,-1,2,89,1,90,1,
1,407,1,1,408,1,1,443,1,1,444,1,2,453,-1,454,-1,1,
453,1,1,261,1,1,262,1,1,457,1,1,458,1,1,9,1,1,10,1,
1,149,1,1,150,1,1,431,-1,1,432,-1,1,165,1,1,166,1,
1,102,1,2,101,-1,102,-1,1,379,1,1,380,1,1,405,1,1,
406,1,1,376,1,2,375,-1,376,-1,1,427,1,1,428,1,1,203,1,
1,204,1,1,449,1,1,450,1,1,459,1,1,460,1,1,39,1,1,
40,1,1,365,1,1,366,1,1,414,1,2,413,-1,414,-1,1,85,1,
1,86,1,1,389,1,1,390,1,1,254,1,2,253,-1,254,-1,1,
411,1,1,412,1,1,384,-1,2,383,1,384,1,1,470,-1,2,469,1,
470,1,1,299,1,1,300,1,1,13,1,1,14,1,2,471,1,472,1,1,
471,-1,1,49,1,1,50,1,2,333,-1,334,-1,1,333,1,1,339,1,
1,340,1,1,302,-1,2,301,1,302,1,1,367,-1,1,368,-1,1,
437,1,1,438,1,1,341,1,1,342,1,1,395,1,1,396,1,1,421,1,
1,422,1,2,473,-1,474,-1,1,473,1,1,153,1,1,154,1,1,
159,1,1,160,1,2,451,1,452,1,1,451,-1,1,448,-1,2,447,1,
448,1,1,71,1,1,72,1,1,455,1,1,456,1,1,175,1,1,176,1,
1,463,1,1,464,1,1,377,1,1,378,1,2,173,-1,174,-1,1,
173,1,1,434,1,2,433,-1,434,-1,2,141,1,142,1,1,141,-1,
1,48,-1,2,47,1,48,1,1,18,1,2,17,-1,18,-1,1,185,1,1,
186,1,2,417,-1,418,-1,1,417,1,2,345,-1,346,-1,1,345,1,
1,249,1,1,250,1,1,285,-1,1,286,-1,1,305,1,1,306,1,1,
59,1,1,60,1,1,189,-1,1,190,-1,2,363,1,364,1,1,363,-1,
1,105,-1,1,106,-1,1,425,1,1,426,1,2,467,-1,468,-1,1,
467,1,1,118,1,2,117,-1,118,-1,1,257,1,1,258,1,1,466,1,
2,465,-1,466,-1,1,279,-1,1,280,-1,1,439,1,1,440,1,1,
273,1,1,274,1,1,95,1,1,96,1,1,401,1,1,402,1,1,55,1,
1,56,1,1,347,-1,1,348,-1,1,99,1,1,100,1,1,399,1,1,
400,1,1,23,1,1,24,1,2,139,1,140,1,1,139,-1,1,205,1,1,
206,1,1,75,1,1,76,1,1,271,1,1,272,1,1,415,1,1,416,1,
1,409,1,1,410,1,1,441,-1,1,442,-1,1,385,1,1,386,1,
2,477,-1,478,-1,1,477,1,1,147,1,1,148,1,1,387,1,1,
388,1,1,129,1,1,130,1,1,479,1,1,480,1,2,281,1,282,1,
1,281,-1,1,393,1,1,394,1,1,303,1,1,304,1,1,197,1,
1,198,1,1,353,1,1,354,1,1,287,1,1,288,1,1,239,1,1,
240,1,1,335,1,1,336,1,1,476,-1,2,475,1,476,1,1,373,1,
1,374,1,1,296,1,2,295,-1,296,-1,1,37,1,1,38,1,1,64,-1,
2,63,1,64,1,1,462,1,2,461,-1,462,-1,1,247,1,1,248,1,
2,325,-1,326,-1,1,325,1,2,309,-1,310,-1,1,309,1
]))>;

return _LR;
