//Standard generators of L2(27) are a and b where a has order 2, b has order 3
//and ab has order 7.
//Standard generators of the double cover 2.L2(27) = SL2(27) are preimages A and
//B where B has order 3 and AB has order 7.  
_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
_LR`AI := [ [a^-1, b^-1], //PSL(2,27).2 = PGL(2,27)
        [a, a^-1*b^-1*a*b*a*b^-1*a*b*a*b*a*b^-1*a*b*a*b^-1*a*b*a*b^-1*a*b]]
                 //PSL(2,27).3 = PSigmaL(2,27)
             where a is (_LR`F).1 where b is (_LR`F).2;
//three constituents, fixed by AI[1], interchanged in 3-cycle by AI[2]
_LR`G :=
MatrixGroup<26, ext<K|Polynomial(K, [-7, 14, -7, 1])> where K is RationalField()
|
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10144821,-21330998,6264728],[
9984226,-14251614,1663953],[-8821218,18203341,
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4123247,-1613991,-1754596],[-13546749,24085391,
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12835294,-20433998,4242639],[
12814664,-26161976,7942137],[1787560,-3662245,
347430],[-9243267,15239215,-3023032],[
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9780056,-15700033,4382100],[583217,2259729,-833082
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8656668,-13693487,2536350],[2875123,-3073108,-69112
],[-3919406,3857149,617823],[
2671505,-5030815,1016521],[7113446,-12670387,
2649316],[-5769555,10809256,-1331169],[
28396337,-49067844,13141271],[
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35623320],[-134907001,242306890,-83016752],[
228590971,-394852587,119767118],[
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295811418,-88436197],[71740682,-129037348,43947045
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431493764,-131555195],[165987133,-299015272,
104455951],[30262693,-42873081,3474926],[
312544904,-541427206,166349128],[
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153875597,-264975026,80789638],[
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1273942],[-2749693,6049822,-3264138],[
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7979365],[16778298,-28888982,8619096],[
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12865201],[24343718,-41996849,12786854],[
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3406340,-5976036,1785668],[
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[-8905450,16416281,-6223022],[
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[9245937,-15171314,3119451],[
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6355651,-10890858,2335037],[
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3103895,-4708418,1104490],[-9801804,16487004,
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11225400,-18129869,4416399],[513874,198308,205436
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4179513],[1177816,-1865888,869542],[
-10284726,16918330,-4223747],[
6964116,-11245459,2550326],[-3454827,4493286,
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-5827248,10199053,-2376682],[-154661,801496,-483058
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11301195,-19804813,4897210],[6172255,-10095328,
2370336],[4531215,-7061476,1750460],[
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1843669,-3322052,783341],[4212655,-7363038,1703420
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8404054,-11572254,3691161],[-15753217,21260424,
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12417871,-16429320,4429365],[
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12632481,-20080079,5926071],[
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3987084,-5862443,1947746],[-7402716,11634025,
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14491014,-21769123,4138829],[
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11085679],[12744423,-20219644,4816874],[
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1732657],[11109339,-16052900,3609581],[
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3486487,-8080185,2647932],[8418540,
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14164351,-24131870,7270105],[-3629066,5864157,
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4677300,-7179932,1977670],[5637364,-7726014,2023301
],[7420575,-12293303,3829348],[
1097692,747609,-509396],[-17041916,26196404,
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10764370,-19484937,6005229],[-2142827,2868551,
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12428324,-18996445,5465060],[
1538531,-1112153,137506],[1305001,-2396548,677163
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4089347,-7515285,2474753],[
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12950311,-23550591,8209126],[
2380403,-3163582,221879],[28086984,-44680137,
13365183],[-2426374,5790180,-2253813],[
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2208346,200084],[11269034,-20143885,6215603],[
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13075136,-22203379,7421129],[
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[5179546,-12417267,4939710],[
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3143412,-9218665,3361675],[
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6030151,-8919285,2518940],[
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22984814,-37912836,9034835],[
7244091,-9587986,1296805],[39131521,-63061491,
14623560],[-9286311,13243791,-2327045],[
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[327786,353484,-1300712],[6609356,
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return _LR;
