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FORMULA
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G.f. = f(x)/g(x) where
f(x) = x^80 - x^79 - x^78 + 2*x^76 + x^74 + x^73 + 2*x^72 + 5*x^70 + 3*x^69 + 13*x^68
+ 10*x^67 + 29*x^66 + 31*x^65 + 66*x^64 + 61*x^63 + 121*x^62 + 138*x^61 +
232*x^60 + 258*x^59 + 396*x^58 + 456*x^57 + 654*x^56 + 739*x^55 + 974*x^54 +
1098*x^53 + 1400*x^52 + 1529*x^51 + 1859*x^50 + 2013*x^49 + 2363*x^48 +
2486*x^47 + 2831*x^46 + 2923*x^45 + 3222*x^44 + 3233*x^43 + 3461*x^42 +
3406*x^41 + 3570*x^40 + 3406*x^39 + 3461*x^38 + 3233*x^37 + 3222*x^36 +
2923*x^35 + 2831*x^34 + 2486*x^33 + 2363*x^32 + 2013*x^31 + 1859*x^30 +
1529*x^29 + 1400*x^28 + 1098*x^27 + 974*x^26 + 739*x^25 + 654*x^24 +
456*x^23 + 396*x^22 + 258*x^21 + 232*x^20 + 138*x^19 + 121*x^18 + 61*x^17 +
66*x^16 + 31*x^15 + 29*x^14 + 10*x^13 + 13*x^12 + 3*x^11 + 5*x^10 + 2*x^8 +
x^7 + x^6 + 2*x^4 - x^2 - x + 1, and
g(x) = x^96 - 2*x^95 - x^94 + 2*x^93 + 2*x^92 -
2*x^90 + x^89 - 3*x^88 + x^87 - 3*x^84 + 4*x^83 + 5*x^82 - 4*x^81 + 4*x^80 -
x^79 - 2*x^78 - 11*x^77 + 5*x^76 - x^75 + 2*x^74 + 5*x^73 + x^72 + 2*x^71 +
8*x^69 - 19*x^68 - 2*x^67 - 2*x^66 + 6*x^65 - 3*x^64 + 10*x^63 + 7*x^62 -
6*x^61 + 11*x^60 - 10*x^59 - 6*x^58 - 13*x^57 + 10*x^56 - 12*x^55 + 6*x^54 +
19*x^53 + 4*x^52 - 3*x^51 - 7*x^50 + 7*x^49 - 20*x^48 + 7*x^47 - 7*x^46 -
3*x^45 + 4*x^44 + 19*x^43 + 6*x^42 - 12*x^41 + 10*x^40 - 13*x^39 - 6*x^38 -
10*x^37 + 11*x^36 - 6*x^35 + 7*x^34 + 10*x^33 - 3*x^32 + 6*x^31 - 2*x^30 -
2*x^29 - 19*x^28 + 8*x^27 + 2*x^25 + x^24 + 5*x^23 + 2*x^22 - x^21 + 5*x^20
- 11*x^19 - 2*x^18 - x^17 + 4*x^16 - 4*x^15 + 5*x^14 + 4*x^13 - 3*x^12 + x^9
- 3*x^8 + x^7 - 2*x^6 + 2*x^4 + 2*x^3 - x^2 - 2*x + 1.
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PROGRAM
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(MAGMA) K:=Rationals(); M:=MatrixAlgebra(K, 4); q1:=DiagonalMatrix(M, [1, -1, 1, -1]); p1:=DiagonalMatrix(M, [1, 1, -1, -1]); q2:=DiagonalMatrix(M, [1, 1, 1, -1]); h:=M![1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1]/2; H:=MatrixGroup<4, K|q1, q2, h, p1>;
(MAGMA, cont.) permstomats:=function(L); n:=#L[1]; M:=MatrixAlgebra(Rationals(), n); a:=#L; MM:=[]; for i in [1..a] do Append(~MM, M ! 0); end for; for i in [1..a] do for j in [1..n] do MM[i][j][L[i][j]]:=1; end for; end for; return MM; end function;
(MAGMA, cont.) MM:=MatrixAlgebra(K, 16); hh:=TensorProduct(M ! 1, h); qq1:=TensorProduct(M ! 1, q1); pp1:=TensorProduct(M ! 1, p1); qq2:=TensorProduct(M ! 1, q2);
(MAGMA, cont.) perm:=sub<Sym(16) | (3, 5)*(4, 6)*(11, 13)*(12, 14), (3, 7)*(4, 8)*(11, 15)*(12, 16), (2, 10)*(4, 12)*(6, 14)*(8, 16), (2, 9)*(4, 11)*(6, 13)*(8, 15)>; Order(perm); pp:=Setseq(Generators(perm)); L:=[Eltseq(pp[1]), Eltseq(pp[2]), Eltseq(pp[3]), Eltseq(pp[4])];
(MAGMA, cont.) ML:=permstomats(L); GG:=MatrixGroup<16, K | hh, qq1, qq2, ML[1], ML[2], ML[3], ML[4]>; Order(GG); MGG:=MolienSeries(GG);
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