/*
2a"M20 = 2^5:A5 as 12 x 12 matrices over Z[b5].
Representation 12b.
Absolutely irreducible representation.
Schur Index 1.

SEED:
Nonzero v fixed by x^2 and y*x*y^-1*x*y*x*y^-1
where <x^2,y*x*y^-1*x*y*x*y^-1> = D8.
v has 2 x 30 = 60 images under G; <v> has 30 images under G.
BASIS:
All in v^G (induced representation).

Possible matrix entries are in {-1,0,1,b5,-b5} (norms 0, 1 only).

Average number of nonzero entries for any element of the group:
19 + 1/5  (19.2; about 13.333%).

  Entry   Av/Mat            %Av/Mat
      0    124.8 [124+4/5]   86.667 [86+2/3]
nonzero     19.2 [ 19+1/5]   13.333 [13+1/3]
     ±1      9.6 [  9+3/5]    6.667 [ 6+2/3]
    ±b5      9.6 [  9+3/5]    6.667 [ 6+2/3]
      1      4.8 [  4+4/5]    3.333 [ 3+1/3]
     -1      4.8 [  4+4/5]    3.333 [ 3+1/3]
     b5      4.8 [  4+4/5]    3.333 [ 3+1/3]
    -b5      4.8 [  4+4/5]    3.333 [ 3+1/3]
*/

R<X>:=PolynomialRing(Rationals());F<b5>:=NumberField(X^2+X-1);r5:=2*b5+1;
// F<r5>:=QuadraticField(5);b5:=(-1+r5)/2;

G<x,y>:=MatrixGroup<12,F|[
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,-b5,-b5,0,0,0,0,0,0,0,0,
0,0,-1,b5,0,0,0,0,0,0,0,0,
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,-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,b5,b5,
0,0,0,0,0,0,0,0,0,0,-b5,1,
0,0,0,0,0,0,0,0,-1,b5,0,0,
0,0,0,0,0,0,0,0,-b5,-b5,0,0]
,[
0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,0,0]
>;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 1625+1000*b5 = (3+b5)^6 (norm 15625 = 5^6).
B1:=MatrixAlgebra(F,12)![
2,b5,0,0,0,0,0,0,0,0,0,0,
b5,2,0,0,0,0,0,0,0,0,0,0,
0,0,2,b5,0,0,0,0,0,0,0,0,
0,0,b5,2,0,0,0,0,0,0,0,0,
0,0,0,0,2,b5,0,0,0,0,0,0,
0,0,0,0,b5,2,0,0,0,0,0,0,
0,0,0,0,0,0,2,b5,0,0,0,0,
0,0,0,0,0,0,b5,2,0,0,0,0,
0,0,0,0,0,0,0,0,2,b5,0,0,
0,0,0,0,0,0,0,0,b5,2,0,0,
0,0,0,0,0,0,0,0,0,0,2,b5,
0,0,0,0,0,0,0,0,0,0,b5,2];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,12)!Eltseq(y^3);