/*
U4(2):2 as 6 x 6 matrices over Z.
Representation 6^+.
Schur index 1.

SEED:
Nonzero v fixed by x and negated by y*x*y^-2 where <x,yxy^-2> = S6 x 2.
v has 2 x 36 = 72 images under G; <v> has 36 images under G.
BASIS:
v, v*y, -v*y*x, v*y*x*y, v*y*x*y^2, v*y*x*y^3. (Note that -v*y*x = v*y^2.)

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

Average number of nonzero entries for any element of the group:
16  (exactly 16; about 44.444%).

Entry   Av/Mat  %Av/Mat
    0    20      55.556 [55+5/9]
   ±1    16      44.444 [44+4/9]
    1     8      22.222 [22+2/9]
   -1     8      22.222 [22+2/9]


LATTICE INFO:
Name: E6.
Minimum: 2.
Kissing number: 72.
Theta series: 1 + 72*q^2 + 270*q^4 + 720*q^6 + 936*q^8 + 2160*q^10 + O(q^12).


p-MODULAR REDUCTIONS:
p = 3: 1^+.5^+.
other: absolutely irreducible.
*/


F:=Rationals();
Z:=Integers();


G<x,y>:=MatrixGroup<6,F|\[
1,0,0,0,0,0,
0,0,-1,0,0,0,
0,-1,0,0,0,0,
0,0,0,1,0,0,
0,0,0,0,1,0,
0,-1,-1,0,0,1]
,\[
0,1,0,0,0,0,
0,0,1,0,0,0,
0,0,0,-1,0,0,
0,0,0,0,1,0,
0,0,0,0,0,1,
1,0,0,-1,0,0]
>;
c:=x;d:=y;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 3; elem divs 1^5.3.
B1:=MatrixAlgebra(F,6)!\[
2,-1,1,1,0,0,
-1,2,-1,-1,1,0,
1,-1,2,1,-1,1,
1,-1,1,2,-1,1,
0,1,-1,-1,2,-1,
0,0,1,1,-1,2];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,6)!1;


LA1:=LatticeWithGram(MatrixAlgebra(Z,6)!B1);
LAD1:=LatticeWithGram(MatrixAlgebra(Z,6)!(3*B1^-1));