/* HN as 133 x 133 matrices over Q(r5). */ F:=QuadraticField(5); // F:=GF(9);r5:=w^2; // F:=GF(11);r5:=F!4; R:=1/r5;S:=2/r5;TF:=F!2/5;FF:=F!4/5;R2:=r5/2;R4:=r5/4;R8:=r5/8; R34:=3*r5/4;R38:=3*r5/8;R58:=5*r5/8;R78:=7*r5/8;R98:=9*r5/8; RR4:=(1+r5)/4;RS4:=(1-r5)/4;RR5:=(1+r5)/5;RS5:=(1-r5)/5; SS4:=(5-r5)/4;SS8:=(5-r5)/8;SR10:=(5+r5)/10;SS10:=(5-r5)/10; SS16:=(5-r5)/16;SR20:=(5+r5)/20;SS20:=(5-r5)/20; SR2:=2*(1+r5)/5;SS5:=(25-5*r5)/16;SS7:=(35-7*r5)/16;RR3:=(5+3*r5)/20; RS3:=(5-3*r5)/20;SS3:=(15-3*r5)/16;SS38:=(15-3*r5)/8; x:=\[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, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-R,-R, -R,R,0,-R,0,0,R,R,0,-R,0,R, -R,0,0,0,0,0,0,0,0,0,0,R,0,0,0,0,0,0,0,0,R, R,0,0,0,0,0,0,0,0,-R,0,0,R,0,-R,0,-R, R,R,R]; print "Have loaded z."; t:=[-R,-SR20,-SR20,SR20, 0,SR20,SR20,0,0,SR20,0,0,-SR20, 0,0,-SR20,0,-SR20,0,0,0,SR20,SR20, 0,-SR20,0,-SR20,0,0,0,SR20,-SR20, 0,0,-SR20,-SR20,0,0,0,0,0,0,0,0,-SR20, SR20,0,0,0,0,0,0,-SR20,0,0,-SR20,0, 0,0,-SR20,SR20,SR20,0,0,0,0,0,-SR20, -SR20,SR20,0,0,0,0,0,0,-SR20,TF,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, -SS20,R,SS20,-SS20,SS20,0,0, SS20,0,0,0,-SS20,0,SS20,SS20,0, -SS20,0,SS20,0,0,-SS20,0,0,0,-SS20, 0,SS20,0,0,0,0,0,0,SS20,0,-SS20,-SS20, 0,SS20,0,SS20,0,-SS20,0,0,-SS20, -SS20,0,-SS20,-SS20,0,SS20,0,0, 0,-SS20,0,0,0,0,0,0,-SS20,0,-SS20,0,0,0, -SS20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,RR5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,RS3, -SR20,0,0,-SR20,-SR10,0,0,0,0,0,0,0,0,0, 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0,0,0,0,0,-SS3,-SS16,0,0,0,-SS16, SS16,0,0,0,0,0,0,SS16,SS16,-SS16, 0,SS16,0,-SS16,0,0,0,-SS16,0,0,0,0,0,0, 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,-RR4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,-SS4,0,SS16,0,SS4,0,SS3,0, SS16,0,-SS4,-SS16,SS4,-SS16, 0,-SS3,0,SS3,0,SS16,-SS3, SS5,-SS4,0,0,-SS4,SS3,SS16, 0,-SS3,-SS4,0,-SS16,SS3, -SS16,-SS3,0,-SS16,0,0,0,0,SS16, -SS4,SS16,-SS3,SS4,0,0,0,-SS16, -SS16,0,SS16,-SS16,-SS3, -SS4,SS4,-SS5,SS16,SS4, -SS3,-SS3,0,-SS4,-SS4,-SS4, 0,SS16,-SS4,0,SS4,0,-SS5, -SS4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-RR4,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,R8,-R58,-R2,R8,0,R8, -R8,R2,-R2,-R2,0,-R4,0,R4,-R8,-R2, -R58,R8,R58,-R2,-R38,-R58,R4,-R34,-R2,0, -R58,R2,R8,R2,-R8,-R8,R8,R38,R8, -R8,-R58,R2,R2,R8,-R8,-R8,-R2,-R38,0, -R2,-R2,R8,-R2,0,0,R8,-R8,-R8,-R38,0, 0,-R2,R4,-R4,0,R8,-R2,-R2,0,-R2,-R34, -R4,R8,R8,-R58,-R8,R34,-R2,-R8,-R58,0, 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,-RS4,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,SS16,SS16,-SS16,0,0,0,0, SS16,0,-SS16,SS16,0,0,0,0,SS16,0, 0,0,0,0,-SS16,SS16,SS16,0,SS16,0, 0,-SS16,0,SS16,0,0,0,SS16,0,-SS16, 0,SS16,0,SS16,-SS3,0,0,0,0,0,SS16, 0,-SS16,-SS16,SS16,0,0,SS16,0, SS16,-SS16,SS16,0,-SS16,0,0,0, SS16,SS16,0,0,-SS16,SS16,0,-SS16, SS16,SS16,SS16,SS16,0,0,0,0,0, 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,-RR4,0,0,0,0,0,0,0,0,0,0,0,0, 0,-R8,0,R2,-R38,0,R38,-R58,R58,R2,-R58, -R8,-R2,0,R2,R2,0,-R2,R8,R2,-R8,0, -R58,-R8,-R38,R8,0,0,R38,0,R2,-R2,R8,0, 0,r5,R2,-R98,-R2,0,R58,0,0,-R8,0,0,0,-R2,0, 0,-R2,0,R2,0,0,0,R58,-R2,-R2,R2,0,-R58,0, R8,-R2,-R8,0,0,-R38,R2,-R8,0,-R8,-R8, 0,0,-R8,R8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-RS4, 0,0,0,0,0,0,0,0,0,0,0,0,-SS16,SS16,0,0, 0,-SS16,0,SS16,0,0,0,0,0,-SS16,0,0, SS16,0,-SS16,-SS16,-SS16,0,0,SS16, SS16,0,0,-SS16,0,0,SS16,SS16, -SS16,0,-SS16,0,0,0,0,0,0,0,-SS16,0,0,0, 0,-SS16,-SS16,0,0,0,0,SS16,0,0,SS16, 0,-SS16,SS16,-SS16,-SS16,SS16, 0,0,0,SS16,SS16,0,-SS16,0,0,-SS16, 0,SS4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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, -RR4,0,0,0,0,0,0,0,0,0,0,0,-R8,0,R8,0,0,0, -R8,R8,R8,R8,-R8,-R8,0,R8,0,0,0,0,0, -R8,0,-R8,0,R8,R8,0,0,0,0,0,0,0,0,-R8,R2, 0,-R8,0,0,R8,0,R8,0,0,-R8,0,-R8,0,0,-R8, -R8,0,0,0,0,R8,0,0,0,0,-R8,0,R8,0,-R8,0,0, 0,0,-R8,0,-R8,-R8,R8,0,0,R8,0,0,0,0,0,0,0, 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,-RS4,0,0,0,0,0,0,0,0,0,0, SS16,0,0,-SS8,0,0,0,-SS16,0,0,SS16, SS16,0,0,SS16,0,0,-SS16,0,0,0,0, -SS16,-SS16,-SS16,SS16,SS16, SS16,0,SS16,0,0,0,0,0,-SS3,0,SS16, 0,-SS16,0,-SS16,SS16,0,SS16, -SS16,SS16,-SS16,0,0,0,0,0,-SS16, -SS16,0,0,-SS16,0,SS16,0,-SS16,0,0, SS16,0,0,0,SS16,SS16,0,SS16,0, -SS16,SS16,SS16,0,0,0,0,0,0,0,0,0,0,0, 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,-RR4,0,0,0,0,0,0,0,0,0,0,-R8, 0,0,0,0,0,0,-R8,0,R8,-R8,0,R8,0,0,-R8,0,0, 0,0,0,R8,-R8,-R8,0,-R8,R8,0,0,0,-R8,R8, 0,0,-R8,0,R8,0,-R8,0,0,0,0,0,0,0,0,0,0,-R2,0, -R8,-R8,0,-R8,0,0,R8,-R8,R8,R8,0,0,0, -R8,-R8,0,0,R8,-R8,0,R8,0,0,0,0,0,0,0,0,0, 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,-RS4,0,0,0,0,0,0,0, 0,0,-R8,0,0,0,0,R8,-R8,0,0,-R8,R8,0,0,R8, -R8,0,-R8,R8,0,0,0,-R8,R8,0,-R8,R8,0, -R8,-R8,0,0,0,R8,-R8,0,0,R8,0,0,0,R8, R8,-R8,0,-R8,0,0,R8,0,-R8,-R8,R8,0, R38,0,-R8,R8,-R8,0,R8,0,-R8,0,0,0,0,0, -R8,0,R8,0,-R8,0,R8,0,0,0,0,0,0,0,0,0,0,0,0, 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,-RS4,0,0,0,0,0,0,0,0,R8, R8,0,-R8,0,-R8,0,R8,0,-R8,0,R8,R8, -R8,R8,R8,R8,0,0,0,-R8,-R8,R8,R4,0, R8,-R8,0,-R8,-R8,R8,0,-R8,0,R8,0,-R8, 0,R8,-R8,0,R8,R8,0,0,0,R8,R8,0,0,0,0,0, R8,R4,-R2,R8,0,0,-R8,-R8,R8,R8,0, R8,R8,0,R8,-R8,R8,-R8,-R8,R8,0,0, R8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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, -RS4,0,0,0,0,0,0,-R2,-R2,0,R38,R2,0, -R8,R58,0,-R8,-R2,-R2,R8,R2,0,-R2, -R2,0,R2,-R58,-R8,-R58,0,-R8,0,-R38,-R2, R8,0,R8,-R8,R8,0,0,R8,-R38,-R58,0,R8, R2,0,R2,-R2,-R2,-R2,-R8,-R58,0,-R8, -R2,-R2,0,0,0,0,R8,-R2,-R8,0,-R38,-R8, R38,0,-R58,-R2,-R2,-R2,0,-R38,0,-R8,-R2, 0,0,0,-R2,R8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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,-RS4,0,0,0,0,0,0,0,-R4,0,-R8,-R8, R8,R8,-R8,-R8,0,R8,R8,-R8,0,0,0,0,0, -R8,-R4,-R8,-R8,R8,0,R8,0,0,-R8,0,0, R8,-R8,R8,-R8,R8,0,R8,R8,-R8,0, -R8,-R8,-R8,0,-R8,0,0,-R8,R8,R8,0,0,0, -R8,0,R8,-R8,-R8,R8,-R8,-R8,0,0,0, -R8,R8,R8,0,-R8,-R8,0,R2,-R8,0,-R8,0, 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,-RS4, 0,0,0,0,R8,0,0,0,0,R8,-R8,0,0,R8,R8, -R8,0,R8,0,R8,-R8,0,0,0,R8,0,R8,-R4, -R8,R8,0,R8,R8,R8,-R8,-R8,R8,R38, R8,0,-R8,0,0,0,0,-R8,0,R8,0,0,0,0,0,0,0,R8, -R8,-R8,R8,0,0,-R8,R4,-R8,0,0,R8,0,0,0, -R8,-R8,R4,R8,-R8,0,R8,0,0,0,0,0,0,0,0,0, 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,-RS4,0,0, 0,0,-R2,-R38,R38,0,0,-R8,R2,-R2,0,0,-R2, R8,R58,-R38,0,-R8,R2,R8,-R2,0,-R58, R38,-R58,0,0,-R38,R8,R2,0,-R8,0,R8,0, R8,-R38,-R8,R2,R2,R8,-R2,0,-R38,0, -R2,-R8,-R2,R2,-R38,0,-R8,0,R8,-R8, -R8,R8,-R8,0,0,-R2,0,R2,0,0,0,-R2,-R2, 0,R8,0,-R2,-R58,R2,0,-R38,-R2,R8,0,0,0,0, 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,-RS4, 0,0,-R8,0,R8,0,0,0,-R8,R8,R8,0,-R8,-R8, 0,R8,0,0,0,0,0,0,0,-R8,0,0,R8,0,0,0,0,0,-R8, 0,0,0,R8,0,-R8,-R8,0,R8,0,R8,0,0,-R8,0, -R8,R8,0,-R8,-R8,0,R8,0,0,R8,-R8,0,0,0, -R8,0,R2,-R8,-R8,0,0,0,0,-R8,0,0,0,R8,0, 0,R8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 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,-RS4,0,0,R8,R8,0,R58,-R8,R38, -R34,0,R58,R2,R8,-R2,-R2,0,R8,R8, -R2,-R58,R58,R8,R98,R38,R8,-R2,0,R4, -R8,0,0,R2,-R2,R8,-R8,-R58,-R2,R58,0, -R58,-R2,R2,0,R8,R8,R8,R2,R34,-R58, R8,0,-R8,-R8,-R2,-R8,0,-r5,R58,R8,0,0, R58,-R8,R8,R8,0,R8,0,R8,-R2,R2,R8, R2,-R8,0,R8,R58,-R38,0,0,0,0,0,0,0,0,0,0,0,0, 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,-RS4]; print "Have loaded t."; G:=MatrixGroup<133,F|x,y,z,t>; print "Have made HN = ."; g1:=x^2; g2:=((y*z)^3*t)^2; print "Have obtained standard generators g1 and g2 (in the sense of R.A.Wilson) for HN.";