/*
S6(2) as 7 x 7 matrices over Q.
Represented using orthogonal matrices.
Absolutely irreducible representation.
Schur index 1.

MONOMIAL GROUP:
M := <z,y> = 2^6:L3(2) where z = (x*y^-2*x*y^2)^(y*x*y*x*y^-1*x).

SEED:
Nonzero vector v fixed by <z,yzy^-1,y^3zy^-3> of order 192.
Stab_G(<v>) = <z,yzy^-1,y^3zy^-3,x> = 2^5:S6 where v*x = -v.
v has 2 x 63 = 126 images under G; <v> has 63 images under G.
BASIS:
v, v*y, v*y^2, v*y^3, v*y^4, v*y^5, v*y^6.

Possible matrix entries are in {-1,-½,0,½,1}.

Average number of nonzero entries for any element of the group:
25 + 2/3  (about 25.667; about 52.381%).


  Entry    Av/Mat               %Av/Mat
      0    23.333  [23 + 1/3]    47.619  [47 + 13/21]
nonzero    25.667  [25 + 2/3]    52.381  [52 +  8/21]
     ±½    24.889  [24 + 8/9]    50.794  [50 + 50/63]
     ±1     0.778  [     7/9]     1.587  [ 1 + 37/63]
      ½    12.444  [12 + 4/9]    25.397  [25 + 25/63]
     -½    12.444  [12 + 4/9]    25.397  [25 + 25/63]
      1     0.389  [    7/18]     0.794  [     50/63]
     -1     0.389  [    7/18]     0.794  [     50/63]


Dbl Coset  #M-cosets    Entries (for any elt in double coset)
                        nonzero   0  ±½  ±1
  M = M1M       1          7     42   0   7
      MxM      14         19     30  16   3
    MxyxM      56         25     24  24   1
  MxyxyxM      64         28     21  28   0


p-MODULAR REDUCTIONS:
p = 2: no reduction possible.
other: absolutely irreducible.
*/


F:=Rationals();

G<x,y>:=MatrixGroup<7,F|[
-1,0,0,0,0,0,0,
0,-1/2,1/2,1/2,0,0,-1/2,
0,1/2,-1/2,1/2,0,0,-1/2,
0,1/2,1/2,-1/2,0,0,-1/2,
0,0,0,0,-1,0,0,
0,0,0,0,0,-1,0,
0,-1/2,-1/2,-1/2,0,0,-1/2]
,\[
0,1,0,0,0,0,0,
0,0,1,0,0,0,0,
0,0,0,1,0,0,0,
0,0,0,0,1,0,0,
0,0,0,0,0,1,0,
0,0,0,0,0,0,1,
1,0,0,0,0,0,0]
>;
a:=x;b:=y;

z:=GL(7,F)!\[
1,0,0,0,0,0,0,
0,1,0,0,0,0,0,
0,0,0,0,0,1,0,
0,0,0,1,0,0,0,
0,0,0,0,0,0,-1,
0,0,1,0,0,0,0,
0,0,0,0,-1,0,0];
z eq (x*y^-2*x*y^2)^(y*x*y*x*y^-1*x);



// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 1.
B1:=MatrixAlgebra(F,7)!1;


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,7)!1;