/*
A6 as 10 x 10 matrices over Z.
Absolutely irreducible representation.
Schur Index 1.

SEED:
Nonzero v negated by x and fixed by y^2*x*y^2*x*y*x*y
where <x,y^2*x*y^2*x*y*x*y> = S4.
v has 2 x 15 = 30 images under G; <v> has 15 images under G.
BASIS:
All in v^G (NSB([x,y,y^-1])+...).

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

Average number of nonzero entries for any element of the group:
20  (20; 20% exactly).

Entry  Av/Mat  %Av/Mat
    0    80       80
   ±1    20       20
    1    10       10
   -1    10       10
*/


F:=Rationals();

G<x,y>:=MatrixGroup<10,F|\[
-1,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,
0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,
0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,0,0,-1]
,\[
0,1,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,
1,-1,0,0,0,1,-1,0,0,0,
0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,
0,1,0,0,-1,0,0,0,1,-1,
1,0,-1,1,0,0,0,0,1,0,
0,0,0,0,0,-1,0,0,0,0,
0,0,-1,0,1,-1,0,1,0,0]
>;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 46656 [e.divs: 1^4.6^6].
B1:=MatrixAlgebra(F,10)!\[
4,1,1,-1,0,-1,1,0,-1,0,
1,4,0,0,1,1,-1,0,-1,1,
1,0,4,1,1,-1,0,1,1,0,
-1,0,1,4,0,0,-1,1,-1,-1,
0,1,1,0,4,1,0,-1,1,-1,
-1,1,-1,0,1,4,1,1,0,0,
1,-1,0,-1,0,1,4,1,0,-1,
0,0,1,1,-1,1,1,4,0,1,
-1,-1,1,-1,1,0,0,0,4,1,
0,1,0,-1,-1,0,-1,1,1,4];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,10)!\[
1,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,1];