/*
A6 as 5 x 5 matrices over Z.
Representation 5a.
Absolutely irreducible representation.
Schur Index 1.

SEED:
Nonzero v fixed by <x,y*x*y^2> = A5.
v has 1 x 6 = 6 images under G; <v> has 6 images under G.
BASIS:
NSB([x,y]) with above v.

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

Average number of nonzero entries for any element of the group:
8 + 1/3  (about 8.333; 33.333%).

Entry  Av/Mat           %Av/Mat
    0  16.667 [16+2/3]   66.667 [66+2/3]
   ±1   8.333  [8+1/3]   33.333 [33+1/3]
    1   4.167  [4+1/6]   16.667 [16+2/3]
   -1   4.167  [4+1/6]   16.667 [16+2/3]
*/


F:=Rationals();

G<x,y>:=MatrixGroup<5,F|\[
1,0,0,0,0,
0,0,1,0,0,
0,1,0,0,0,
0,0,0,0,1,
0,0,0,1,0]
,\[
0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
1,0,0,0,0,
-1,-1,-1,-1,-1]
>;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 1296 [e.divs: 1.6^4].
B1:=MatrixAlgebra(F,5)!\[
5,-1,-1,-1,-1,
-1,5,-1,-1,-1,
-1,-1,5,-1,-1,
-1,-1,-1,5,-1,
-1,-1,-1,-1,5];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,5)!\[
1,0,0,0,0,
0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
0,0,0,0,1];