_LR := rec < recformat< F: GrpFP, AI: SeqEnum, G: GrpMat > |
      F := FreeGroup(2) >;
//Standard generators of O8+(2) are a and b where a is in class 2E, b is in
//class 5A, ab is in class 12F (or 12G) and ababababbababbabb has order 8.
//[Actually, b is allowed to be in classes 5B or 5C as they are automorphic to
//class 5A, but then you must change the class you allow ab to belong to. To
//ensure that ab is in the correct class relative to the class of b, we just
//need to check that abb has order 15.]
//Standard generators of the double cover 2.O8+(2) are preimages A and B where
//B has order 5, ABB has order 15 and B is in class +5A.
//[In 2.O8+(2), class +5A is not automorphic to classes +5B or +5C. A
//condition equivalent to B being in class +5A is that ABABB has order 15.]

_LR`AI := [ [a, b*a*b^-1*a*b^-1*a*b^2*a*b*a*b*a*b^-1], //order 2
            [a, b^-1*a*b^2*a*b^2*a*b^-2*a*b^2*a*b*a*b] ] //order 3
                    where a is (_LR`F).1 where b is (_LR`F).2;

//irreducible, fixed by AI[1], mapped to 2 other reps by AI[2].
_LR`G := sub<GL(84,Integers()) |
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0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
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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,
0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,
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0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,
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0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,0,
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1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,-1,0,0,0,0,
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0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,
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,
1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,
0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
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,
1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,
0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,
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0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
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0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,1,0,-1,0,0,1,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,-1,1,-1,
0,0,0,-1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,
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,
0,1,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,-1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,1,0,0,0,0,0,0,-1,0,
0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,
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,-1,0,0,0,1,0,-1,0,0,0,0,
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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,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
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0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
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0,0,0,0,-1,0,0,1,0,-1,0,0,0,1,0,1,0,0,0,0,0,0,-1,0,
0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,
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0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,
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,
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,
0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,2,1,0,0,
0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
0,0,-1,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,
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0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,
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0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,1,0,0,0,0,
0,1,0,1,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,
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0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,1,0,
0,0,0,0,0,-1,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
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1,0,-1,0,0,-1,0,-1,0,0,0,0,0,0,1,0,0,-1,-1,0,0,0,0,0,
0,0,0,0,0,0,0,1,1,-1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,
0,0,1,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
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,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,1,0,0,0,
0,1,0,0,0,0,0,0,-1,0,0,2,1,0,0,0,0,1,0,0,0,0,1,0,0,
-1,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,1,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
1,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,
1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,1,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,
0,0,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,
0,0,0,-1,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,-1,
0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,
-1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,
0,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,
0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,
0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,-1,0,1,0,
0,1,0,1,0,0,0,0,-1,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,
0,1,0,0,0,0,0,-1,-1,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,
0,1,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,1,0,0,-1,0,0,0,
1,0,-1,0,0,-1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,
0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,
0,0,1,1,0,0,-1,0,0,0,1,0,-1,0,0,-1,0,0,2,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,
0,-1,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1,0,1,0,
-1,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,
-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,1,0,
0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,-1,0,0,0,0,0,0,1,-1,-1,0,0,1,1,0,1,0,0,-1,0,0,0,0,
0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,
0,0,-1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,0,0,1,
1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,
0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,1,0,-1,0,0,0,1,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,
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,-1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,
-1,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,
-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,1,
0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,-1,0,0,0,1,0,0,1,0,-1,0,0,2,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,1,
1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,-1,0,-1,0,-1,0,0,0,
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,
0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,-1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,-1,-1,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,-1,1,0,0,0,1,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,
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,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,
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,
1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,1,0,
0,0,0,0,0,1,0,-1,0,0,-1,0,0,2,-1,0,-1,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,
0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
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,
0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,-1,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,-1,0,0,0,0,-1,0,1,0,0,0,0,1,0,0,0,0,0,0,-1,
0,0,1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,-1,0,0,-1,-1,
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,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,
-2,-1,0,0,0,-1,0,0,0,0,1,0,1,0,-1,0,0,0,0,-1,0,0,0,1,
0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,-1,0,0,0,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,
-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,
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,
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,-2,-1,0,0,
0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,
0,0,0,1,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,0,0,0,0,0,0,0,
0,0,1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,
1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,
0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,-1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
-1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,
0,0,0,0,0,0,1,0,0,-1,-1,0,0,-1,0,0,0,0,1,0,0,0,0,0,
0,0,0,-1,-2,-1,0,0,0,-1,0,1,0,0,1,0,1,0,0,0,0,-1,0,0,
0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,-1,
1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,0,0,0,0,-1,-1,0,
0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,
0,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0 ] >;

return _LR;
