/*
A6 as 8 x 8 matrices over Z[b5].
Representation 8b.
Absolutely irreducible representation.
Schur Index 1.

SEED:
Nonzero v fixed by <x,y*x*y^-1> = D8.
v has 1 x 45 = 45 images under G; <v> has 45 images under G.
BASIS:
All in v^G.

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

Average number of nonzero entries for any element of the group:
31 + 13/45  (about 31.289; 48.889%).

  Entry  Av/Mat             %Av/Mat
      0  32.711 [32+32/45]   51.111 [51+1/9]
nonzero  31.289 [31+13/45]   48.889 [48+8/9]
     ±1  15.644 [15+29/45]   24.444 [24+4/9]
      1   7.822  [7+37/45]   12.222 [12+2/9]
     -1   7.822  [7+37/45]   12.222 [12+2/9]
  ±1±b5  12.644 [15+29/45]   24.444 [24+4/9]
     b5   3.911  [3+41/45]    6.111  [6+1/9]
    -b5   3.911  [3+41/45]    6.111  [6+1/9]
   1+b5   3.911  [3+41/45]    6.111  [6+1/9]
  -1-b5   3.911  [3+41/45]    6.111  [6+1/9]
*/


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<8,F|[
1,0,0,0,0,0,0,0,
1,-1,1,1,1,1,-1,1,
0,0,0,0,0,0,0,1,
b5+1,0,0,b5,b5+1,b5,-b5-1,0,
-b5-1,0,0,-b5,-b5-1,-b5,b5,0,
0,0,0,0,0,1,0,0,
0,0,0,-1,-1,0,0,0,
0,0,1,0,0,0,0,0]
,[
1,0,1,0,0,1,0,1,
1,0,0,0,b5+1,0,-b5-1,0,
0,0,0,0,1,0,0,0,
b5+1,0,0,0,1,0,-b5-1,0,
-b5-1,0,0,b5,-1,-1,1,0,
0,-1,0,0,0,0,-1,0,
0,b5,0,0,-b5,0,0,1,
0,0,0,1,0,0,0,0]
>;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 59049 = 3^10.
B1:=MatrixAlgebra(F,8)![
8,-3*b5-1,-1,-1,-3*b5-1,-4,3*b5+2,-1,
-3*b5-1,8,3*b5+2,3*b5+2,-3*b5-1,3*b5+2,-4,-3*b5-1,
-1,3*b5+2,8,-3*b5-1,-1,-1,-3*b5-1,-4,
-1,3*b5+2,-3*b5-1,8,-4,3*b5+2,-3*b5-1,-1,
-3*b5-1,-3*b5-1,-1,-4,8,-1,3*b5+2,3*b5+2,
-4,3*b5+2,-1,3*b5+2,-1,8,-3*b5-1,-1,
3*b5+2,-4,-3*b5-1,-3*b5-1,3*b5+2,-3*b5-1,8,3*b5+2,
-1,-3*b5-1,-4,-1,3*b5+2,-1,3*b5+2,8];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,8)![
1,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,
0,0,0,1,0,0,0,0,
0,0,0,0,1,0,0,0,
0,0,0,0,0,1,0,0,
0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,1];