//L3(9) not on online ATLAS.
//I have taken standard generators of U3(9) as a and b where a has order 2,
//b is in class 3B and ab has order 24.
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
   _LR`AI:=[ [a, b^-1], //duality
  [a, (b*a)^6 * (b^-1*a)^6 * b^-1], //galois
  [a, (b^-1*a)^6 * (b*a)^6 * b]]
                  where a is (_LR`F).1 where b is (_LR`F).2;
//4 constituents
_LR`G :=
/*
Original group: c9Group("l39p")
Direct induction from degree 1
Schur index: 1
Character: ( 364, -36, 40, 4, -4, -4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 
-4, 4, 4, 4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, -4, -4, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
*/

MatrixGroup<364,IntegerRing() |
Matrix(SparseMatrix(364,364,\[
1,5,1,1,6,1,1,7,1,1,8,1,1,1,1,1,2,1,1,3,1,1,4,1,
1,9,-1,1,10,-1,1,11,-1,1,12,-1,1,25,1,1,26,1,1,27,1,
1,28,1,1,29,1,1,30,-1,1,31,1,1,32,1,1,33,1,1,34,-1,
1,35,1,1,36,1,1,13,1,1,14,1,1,15,1,1,16,1,1,17,1,1,
18,-1,1,19,1,1,20,1,1,21,1,1,22,-1,1,23,1,1,24,1,1,
49,1,1,50,1,1,51,1,1,52,1,1,53,1,1,54,-1,1,55,1,1,
56,1,1,57,1,1,58,1,1,59,1,1,60,1,1,37,1,1,38,1,1,
39,1,1,40,1,1,41,1,1,42,-1,1,43,1,1,44,1,1,45,1,1,
46,1,1,47,1,1,48,1,1,85,1,1,86,1,1,87,1,1,88,1,1,
89,1,1,90,1,1,91,-1,1,92,1,1,93,1,1,94,1,1,95,1,1,
96,1,1,97,1,1,98,1,1,99,1,1,100,1,1,101,1,1,102,1,
1,103,1,1,104,1,1,105,1,1,106,1,1,107,1,1,108,1,1,
61,1,1,62,1,1,63,1,1,64,1,1,65,1,1,66,1,1,67,-1,1,
68,1,1,69,1,1,70,1,1,71,1,1,72,1,1,73,1,1,74,1,1,
75,1,1,76,1,1,77,1,1,78,1,1,79,1,1,80,1,1,81,1,1,
82,1,1,83,1,1,84,1,1,183,1,1,184,-1,1,182,1,1,181,1,
1,185,1,1,186,1,1,187,1,1,188,1,1,117,-1,1,118,-1,1,
119,-1,1,120,-1,1,217,1,1,218,1,1,219,-1,1,220,1,1,
225,1,1,226,-1,1,227,1,1,228,1,1,237,-1,1,238,1,1,
239,1,1,240,-1,1,133,-1,1,134,-1,1,135,-1,1,136,-1,
1,253,1,1,254,1,1,255,1,1,256,1,1,157,1,1,158,1,
1,159,1,1,160,1,1,145,1,1,146,1,1,147,1,1,148,1,
1,277,-1,1,278,1,1,279,1,1,280,1,1,207,-1,1,208,-1,
1,206,1,1,205,1,1,141,1,1,142,1,1,143,1,1,144,1,1,
161,-1,1,162,-1,1,163,-1,1,164,-1,1,165,-1,1,166,-1,
1,167,-1,1,168,-1,1,251,-1,1,252,1,1,250,1,1,249,1,
1,222,1,1,221,1,1,224,1,1,223,1,1,301,1,1,302,1,
1,303,-1,1,304,1,1,112,1,1,111,1,1,109,1,1,110,-1,
1,113,1,1,114,1,1,115,1,1,116,1,1,317,1,1,318,1,1,
319,1,1,320,-1,1,243,1,1,244,-1,1,242,1,1,241,1,1,
296,1,1,295,1,1,293,-1,1,294,-1,1,265,1,1,266,1,1,
267,1,1,268,1,1,156,1,1,155,1,1,153,-1,1,154,-1,1,
309,1,1,310,-1,1,311,1,1,312,1,1,246,-1,1,245,1,1,
248,1,1,247,-1,1,121,1,1,122,1,1,123,-1,1,124,1,1,
174,1,1,173,1,1,176,1,1,175,1,1,125,1,1,126,-1,1,
127,1,1,128,1,1,272,1,1,271,1,1,269,-1,1,270,1,1,
341,-1,1,342,1,1,343,1,1,344,-1,1,129,-1,1,130,1,1,
131,1,1,132,-1,1,196,1,1,195,1,1,193,1,1,194,-1,1,
214,1,1,213,-1,1,216,-1,1,215,1,1,172,1,1,171,1,1,
169,-1,1,170,1,1,137,1,1,138,1,1,139,1,1,140,1,1,
284,1,1,283,-1,1,281,-1,1,282,-1,1,274,-1,1,273,1,
1,276,1,1,275,1,1,201,1,1,202,1,1,203,1,1,204,1,1,
231,-1,1,232,1,1,230,1,1,229,1,1,262,1,1,261,-1,1,
264,1,1,263,1,1,149,-1,1,150,1,1,151,1,1,152,1,1,
259,-1,1,260,-1,1,258,-1,1,257,1,1,289,-1,1,290,1,1,
291,1,1,292,1,1,285,-1,1,286,1,1,287,1,1,288,1,1,
199,-1,1,200,-1,1,198,1,1,197,1,1,361,1,1,362,1,1,
363,-1,1,364,1,1,177,1,1,178,1,1,179,-1,1,180,1,1,
305,-1,1,306,-1,1,307,-1,1,308,-1,1,209,1,1,210,-1,
1,211,1,1,212,1,1,313,-1,1,314,-1,1,315,-1,1,316,-1,
1,189,1,1,190,1,1,191,1,1,192,-1,1,331,1,1,332,-1,1,
330,1,1,329,1,1,325,-1,1,326,-1,1,327,-1,1,328,-1,1,
324,1,1,323,1,1,321,1,1,322,-1,1,340,1,1,339,-1,1,
337,-1,1,338,-1,1,335,-1,1,336,-1,1,334,-1,1,333,1,
1,233,-1,1,234,1,1,235,1,1,236,-1,1,345,-1,1,346,-1,
1,347,-1,1,348,-1,1,353,1,1,354,1,1,355,-1,1,356,1,
1,349,1,1,350,1,1,351,-1,1,352,1,1,357,-1,1,358,-1,
1,359,-1,1,360,-1,1,297,1,1,298,1,1,299,-1,1,300,1
])),Matrix(SparseMatrix(364,364,\[
1,9,1,1,10,1,1,11,1,1,12,1,1,17,1,1,18,1,1,19,1,
1,20,1,1,13,1,1,14,1,1,15,1,1,16,1,1,1,1,1,2,1,1,
3,1,1,4,1,1,21,1,1,22,1,1,23,1,1,24,1,1,5,1,1,6,1,
1,7,1,1,8,1,1,37,1,1,38,1,1,39,1,1,40,1,1,28,1,1,
27,1,1,25,1,1,26,1,1,41,1,1,42,-1,1,43,1,1,44,1,1,
31,1,1,32,1,1,30,1,1,29,1,1,45,1,1,46,-1,1,47,1,1,
48,1,1,33,1,1,34,1,1,35,1,1,36,1,1,61,1,1,62,1,1,
63,1,1,64,1,1,69,1,1,70,1,1,71,1,1,72,1,1,77,1,1,
78,1,1,79,1,1,80,1,1,65,1,1,66,1,1,67,1,1,68,1,1,
49,1,1,50,1,1,51,1,1,52,1,1,73,1,1,74,1,1,75,1,1,
76,-1,1,53,1,1,54,1,1,55,1,1,56,-1,1,81,1,1,82,1,1,
83,1,1,84,-1,1,57,1,1,58,1,1,59,1,1,60,-1,1,109,1,
1,110,1,1,111,-1,1,112,1,1,117,1,1,118,1,1,119,1,
1,120,1,1,193,1,1,194,1,1,195,1,1,196,1,1,136,1,
1,135,1,1,133,1,1,134,1,1,201,1,1,202,-1,1,203,1,
1,204,1,1,205,1,1,206,-1,1,207,1,1,208,-1,1,113,1,
1,114,1,1,115,-1,1,116,1,1,85,1,1,86,1,1,87,1,1,
88,1,1,121,1,1,122,1,1,123,1,1,124,1,1,89,1,1,90,1,
1,91,1,1,92,1,1,229,1,1,230,1,1,231,-1,1,232,1,1,
241,1,1,242,1,1,243,1,1,244,1,1,173,1,1,174,-1,1,
175,1,1,176,1,1,93,1,1,94,-1,1,95,1,1,96,1,1,177,1,
1,178,1,1,179,1,1,180,1,1,265,1,1,266,1,1,267,1,1,
268,1,1,281,1,1,282,1,1,283,1,1,284,-1,1,285,-1,1,
286,-1,1,287,1,1,288,1,1,289,-1,1,290,1,1,291,1,1,
292,1,1,154,1,1,153,1,1,156,1,1,155,-1,1,107,1,1,
108,-1,1,106,-1,1,105,1,1,293,1,1,294,1,1,295,1,1,
296,1,1,99,1,1,100,-1,1,98,1,1,97,1,1,261,1,1,262,1,
1,263,1,1,264,1,1,300,1,1,299,-1,1,297,1,1,298,1,1,
309,1,1,310,1,1,311,1,1,312,1,1,270,1,1,269,-1,1,
272,1,1,271,1,1,137,1,1,138,-1,1,139,1,1,140,1,1,
325,1,1,326,-1,1,327,1,1,328,1,1,323,1,1,324,-1,1,
322,1,1,321,-1,1,168,1,1,167,1,1,165,1,1,166,1,1,
159,1,1,160,1,1,158,1,1,157,-1,1,213,1,1,214,1,1,
215,1,1,216,1,1,333,1,1,334,-1,1,335,1,1,336,-1,1,
258,1,1,257,1,1,260,1,1,259,-1,1,222,-1,1,221,1,1,
224,-1,1,223,-1,1,233,1,1,234,-1,1,235,-1,1,236,1,
1,125,1,1,126,-1,1,127,1,1,128,1,1,332,1,1,331,1,
1,329,1,1,330,-1,1,245,1,1,246,1,1,247,1,1,248,1,
1,129,1,1,130,1,1,131,1,1,132,1,1,199,1,1,200,1,
1,198,-1,1,197,-1,1,172,1,1,171,1,1,169,1,1,170,1,
1,225,-1,1,226,1,1,227,1,1,228,-1,1,141,1,1,142,1,
1,143,1,1,144,1,1,273,1,1,274,1,1,275,1,1,276,1,1,
304,-1,1,303,1,1,301,1,1,302,-1,1,145,1,1,146,1,1,
147,1,1,148,1,1,150,1,1,149,1,1,152,-1,1,151,1,1,
278,1,1,277,1,1,280,1,1,279,1,1,162,-1,1,161,-1,1,
164,-1,1,163,1,1,212,1,1,211,1,1,209,1,1,210,1,1,
255,1,1,256,1,1,254,1,1,253,1,1,307,1,1,308,1,1,
306,1,1,305,-1,1,192,1,1,191,-1,1,189,1,1,190,1,1,
181,-1,1,182,-1,1,183,1,1,184,1,1,313,1,1,314,1,1,
315,1,1,316,1,1,185,1,1,186,1,1,187,1,1,188,1,1,
361,1,1,362,-1,1,363,-1,1,364,-1,1,104,-1,1,103,1,
1,101,1,1,102,1,1,252,-1,1,251,1,1,249,1,1,250,1,
1,351,1,1,352,-1,1,350,1,1,349,1,1,337,1,1,338,1,
1,339,1,1,340,1,1,217,1,1,218,-1,1,219,1,1,220,-1,
1,317,-1,1,318,-1,1,319,1,1,320,1,1,357,1,1,358,1,
1,359,1,1,360,1,1,237,1,1,238,1,1,239,1,1,240,1,
1,345,1,1,346,1,1,347,1,1,348,1,1,353,1,1,354,1,1,
355,1,1,356,1,1,341,-1,1,342,1,1,343,-1,1,344,-1
]))>;

return _LR;
