%I #12 Sep 08 2022 08:45:17
%S 1,2,4,8,20,42,95,201,465,1014,2292,5014,11043,23594,49910,102699,
%T 207520,408597,788081,1484174,2738264,4944037,8755686,15208359,
%U 25952790,43527033,71841425,116752322,187018934,295451249,460722893,709561513,1080061323
%N Molien series for a certain 16-dimensional group of order 10321920.
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%p u2:= (1-t^2)*(1-t^4)^2*(1-t^6)^2*(1-t^8)^4*(1-t^10)^2* (1-t^12)^2*(1-t^14)*(1-t^28)*(1-t^30); u2:=subs(t=sqrt(t),u2);
%p u1:= 1 + t + 3*t^4 + 9*t^5 + 24*t^6 + 46*t^ 7 + 117*t^8 + 239*t^9 + 541*t^10 + 1133 *t^11 + 2370*t^12 + 4649*t^13 + 8923*t^14 + 16245*t^15 + 28601*t^16 + 48132 *t^17 + 78194 *t^18 +121981 *t^19 +183920 *t^20 +267517 *t^21 + 376916 *t^22 + 514682 *t^23 + 683056*t^24 + 881972*t^25 + 1110910*t^26 +1366468 *t^27 + 1644918*t^28 + 1940048*t^29 + 2245177*t^30 + 2551867*t^31 + 2851403*t^32 + 3133830 *t^33 + 3389363*t^34 + 3608201 *t^35 + 3781448*t^36 + 3901399*t^37 + 3962896*t^38;
%p u1a:=expand(t^77*subs(t=1/t,u1)); u3:=(u1+u1a)/u2;
%o (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>;
%o 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;
%o MM:=MatrixAlgebra(K,16); hh:=TensorProduct(M ! 1,h); qq1:=TensorProduct(M ! 1,q1); pp1:=TensorProduct(M ! 1,p1); qq2:=TensorProduct(M ! 1,q2);
%o 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);
%o pp:=Setseq(Generators(perm)); L:=[Eltseq(pp[1]),Eltseq(pp[2]),Eltseq(pp[3]),Eltseq(pp[4])]; ML:=permstomats(L); UU:=MatrixGroup<16,K | hh,qq2,ML[1],ML[2],ML[3],ML[4]>; Order(UU); MUU:=MolienSeries(UU);
%K nonn
%O 0,2
%A _N. J. A. Sloane_ and Gabriele Nebe, Apr 26 2005