/*
2"A5 as 8 x 8 matrices over Z.
Irreducible over Q, becomes 4 + 4 over C.
Schur Index 2 (for the irreducible constituents).

SEED:
Undefined.
v [= e1] has 2 x 20 = 40 images under G; <v> has 20 images under G.
BASIS:
v, v.x, v.y, v.x*y, v.y^2, v.x*y^2, v.x^3*y*x*y^2*x, v.y*x*y^2*x.

Possible matrix entries for any element of the group:
The 3 elements of {-1,0,1} only.

Average number of nonzero entries for any element of the group:
32 (50% exactly).

Entry    Av/Mat  %Av/Mat
 0        32      50
 1        16      25
-1        16      25
nonzero   32      50
*/


F:=Rationals();

G<x,y>:=MatrixGroup<8,F|\[
0,1,0,0,0,0,0,0,
-1,0,0,0,0,0,0,0,
-1,0,0,1,-1,1,-1,-1,
0,-1,-1,0,-1,-1,1,-1,
1,0,-1,1,-1,-1,0,-1,
0,1,-1,-1,1,-1,1,0,
-1,0,-1,-1,0,-1,1,-1,
0,-1,1,-1,1,0,1,1]
,\[
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,
1,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,1]
>;

A:=x;B:=y;


// Forms: B1 (Symmetric); B2,B3,B4 (Antisymmetric).
// The antisymmetric forms span a 3-space.
// a*B2+b*B3+c*B4 has determinant (a^2-3*a*b+3*b^2+3*c^2)^4.

// B1 (Symmetric form): Determinant 1.
B1:=MatrixAlgebra(F,8)!\[
2,0,0,-1,0,1,0,-1,
0,2,1,0,-1,0,1,0,
0,1,2,0,0,-1,0,-1,
-1,0,0,2,1,0,1,0,
0,-1,0,1,2,0,0,-1,
1,0,-1,0,0,2,1,0,
0,1,0,1,0,1,2,0,
-1,0,-1,0,-1,0,0,2];

// B2 (Antisymmetric form): Determinant 1.
B2:=MatrixAlgebra(F,8)!\[
0,1,0,0,0,0,0,0,
-1,0,0,1,0,-1,0,1,
0,0,0,1,0,0,0,0,
0,-1,-1,0,0,1,0,1,
0,0,0,0,0,1,0,0,
0,1,0,-1,-1,0,0,1,
0,0,0,0,0,0,0,1,
0,-1,0,-1,0,-1,-1,0];

// B3 (Antisymmetric form): Determinant 81.
B3:=MatrixAlgebra(F,8)!\[
0,0,1,0,-1,0,1,0,
0,0,0,-1,0,1,0,-1,
-1,0,0,0,1,0,1,0,
0,1,0,0,0,-1,0,-1,
1,0,-1,0,0,0,1,0,
0,-1,0,1,0,0,0,-1,
-1,0,-1,0,-1,0,0,0,
0,1,0,1,0,1,0,0];

// B4 (Antisymmetric form): Determinant 81.
B4:=MatrixAlgebra(F,8)!\[
0,0,0,1,0,-1,0,1,
0,0,1,0,-1,0,1,0,
0,-1,0,0,0,1,0,1,
-1,0,0,0,1,0,1,0,
0,1,0,-1,0,0,0,1,
1,0,-1,0,0,0,1,0,
0,-1,0,-1,0,-1,0,0,
-1,0,-1,0,-1,0,0,0];

BS1:=B2;BS2:=B2+B3;BS3:=B4;
// a*BS1+b*BS1+c*BS3 has determinant (a^2-a*b+b^2+3*c^2)^4.


// Centralising algebra: 4-dimensional.
// C1,C2,C3,C4 correspond respectively to the quaternions 1,i,r3*j,r3*k.

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];

C2:=MatrixAlgebra(F,8)!\[
0,1,0,0,0,0,0,0,
-1,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,
0,0,-1,0,0,0,0,0,
0,0,0,0,0,1,0,0,
0,0,0,0,-1,0,0,0,
0,0,0,0,0,0,0,1,
0,0,0,0,0,0,-1,0];

C3:=MatrixAlgebra(F,8)!\[
0,3,-2,0,2,0,-2,0,
3,0,0,2,0,-2,0,2,
2,0,0,3,-2,0,-2,0,
0,-2,3,0,0,2,0,2,
-2,0,2,0,0,3,-2,0,
0,2,0,-2,3,0,0,2,
2,0,2,0,2,0,0,3,
0,-2,0,-2,0,-2,3,0];

C4:=MatrixAlgebra(F,8)!\[
3,0,0,2,0,-2,0,2,
0,-3,2,0,-2,0,2,0,
0,-2,3,0,0,2,0,2,
-2,0,0,-3,2,0,2,0,
0,2,0,-2,3,0,0,2,
2,0,-2,0,0,-3,2,0,
0,-2,0,-2,0,-2,3,0,
-2,0,-2,0,-2,0,0,-3];