/*
2"A5 as 4 x 4 matrices over Z[i6].
Absolutely irreducible representation.
Schur Index 2.

SEED:
Nonzero v s.t. v.(1+i6*x+y+y^2) = 0. v is fixed by <y^x> = 3.
v has 2 x 20 = 40 images under G; <v> has 20 images under G.
BASIS:
v, v.x, v.y, v.y^2*x.

Possible matrix entries for any element of the group:
The 31 elements of {0,±1,±2,±4,±i6,±i6±1,±i6±2,±i6±3,±i6±5,±2i6,±2i6±4} only.
The possible norms are in {0,1,4,6,7,10,15,16,24,31,40} only.

Average number of nonzero entries for any element of the group:
13 + 1/5  (13.2; 82.5% exactly).

    Entry         Av/Mat  %Av/Mat
     0             2.8     17.5
    ±1             2.0 ea  12.5  each
    ±2             0.4 ea   2.5  each
    ±4             0.6 ea   3.75 each
 ±i6               0.4 ea   2.5  each
 ±i6±1             0.6 ea   3.75 each
 ±i6±2             0.2 ea   1.25 each
 ±i6±3             0.2 ea   1.25 each
 ±i6±5             0.2 ea   1.25 each
±2i6               0.4 ea   2.5  each
±2i6±4             0.2 ea   1.25 each
Norm 0 [0]         2.8     17.5
Norm 1 [±1]        4.0     25
Norm 4 [±2]        0.8      5
Norm 6 [±i6]       0.8      5
Norm 7 [±i6±1]     2.4     15
Norm 10 [±i6±2]    0.8      5
Norm 15 [±i6±3]    0.8      5
Norm 16 [±4]       1.2      7.5
Norm 24 [±2i6]     0.8      5
Norm 31 [±i6±5]    0.8      5
Norm 40 [±2i6±4]   0.8      5
nonzero           13.2     82.5
*/


F<i6>:=QuadraticField(-6);

G<x,y>:=MatrixGroup<4,F|[
0,1,0,0,
-1,0,0,0,
i6,-1,0,-1,
1,i6,1,0]
,[
0,0,1,0,
0,1,0,0,
-1,-i6,-1,0,
-1,-2,i6+1,1]
>;

A:=x;B:=y;

// Complex conjugates of x and y.
xc:=GL(4,F)![Conjugate(u):u in Eltseq(x)];
yc:=GL(4,F)![Conjugate(u):u in Eltseq(y)];

// Forms: B1 (Antisymmetric); B2 (Hermitian).

// B1 (Antisymmetric form): Determinant 1.
B1:=MatrixAlgebra(F,4)!\[
0,0,1,0,
0,0,0,-1,
-1,0,0,-1,
0,1,1,0];

// B2 (Hermitian form): Determinant 36.
B2:=MatrixAlgebra(F,4)![
12,-4*i6,i6+6,-4*i6,
4*i6,12,4*i6,-i6+6,
-i6+6,-4*i6,12,-i6-6,
4*i6,i6+6,i6-6,12];

// B2a = -B2/i6: Determinant 1.
B2a:=MatrixAlgebra(F,4)![
2*i6,4,i6-1,4,
-4,2*i6,-4,i6+1,
i6+1,4,2*i6,-i6+1,
-4,i6-1,-i6-1,2*i6];

// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,4)!\[
1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1];