/*
2a"M20 = 2^5:A5 as 10 x 10 monomial matrices over Z.
Representation 10b.
Absolutely irreducible representation.
Schur Index 1.

SEED:
Nonzero v negated by x and fixed by y*x*y*x*y^-1*x*y^-1
where <x,y*x*y*x*y^-1*x*y^-1> = 2^5:S3.
v has 2 x 10 = 20 images under G; <v> has 10 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:
10  (exactly 10%).

Entry  Av/Mat  %Av/Mat
    0      90       90
   ±1      10       10
    1       5        5
   -1       5        5
*/

F:=Rationals();

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


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 1 [e.divs: 1^6].
B1:=MatrixAlgebra(F,10)!Eltseq(y^3);


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,10)!Eltseq(y^3);