/*
S6(2) as 7 x 7 matrices over Z.
Absolutely irreducible representation.
Schur index 1.
Integrally equivalent to representation in S62G1-Zr7B0.M, but with
sparser generators.

SEED:
Nonzero vector v such that v*x = v*y*x*y^-1 = v*y^2*x*y^-2 = v*y^3*x*y^-3 =
v*y^4*x*y^-4 = v*y^5*x*y^-5 = -v
where <x,y*x*y^-1,y^2*x*y^-2,y^3*x*y^-3,y^4*x*y^-4,y^5*x*y^-5> = U4(2):2.
v has 2 x 28 = 56 images under G; <v> has 28 images under G.
BASIS:
v, v*y, v*y^2, v*y^3, v*y^4, v*y^5, v*y^6.

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

Average number of nonzero entries for any element of the group:
21  (21; about 42.857%).

Entry    Av/Mat         %Av/Mat
    0    28   [28]       57.143 [57+1/7]
   ±1    21   [21]       42.857 [42+6/7]
    1    10.5 [10+1/2]   21.429 [21+3/7]
   -1    10.5 [10+1/2]   21.429 [21+3/7]


p-MODULAR REDUCTIONS:
p = 2: 6.1 (uniserial).
other: absolutely irreducible.


LATTICE DETAILS:
Name: E7*.
Aut grp: 2 x S6(2).
Minimum: 3.
Kiss no: 56.
Determinant: 64.
Elmnty divs: 1.2^6.

Theta series:
1 + 56*q^3 + 126*q^4 + 576*q^7 + 756*q^8 + 1512*q^11 + 2072*q^12 + 4032*q^15 +
4158*q^16 + 5544*q^19 + 7560*q^20 + 12096*q^23 + 11592*q^24 + 13664*q^27 +
16704*q^28 + O(q^31).

Minimum vectors are all in v^G.
*/


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

G<x,y>:=MatrixGroup<7,F|\[
-1,0,0,0,0,0,0,
0,-1,0,0,0,0,0,
0,0,-1,0,0,0,0,
0,0,0,-1,0,0,0,
0,0,0,0,-1,0,0,
0,0,0,0,0,-1,0,
1,0,-1,-1,0,1,1]
,\[
0,1,0,0,0,0,0,
0,0,1,0,0,0,0,
0,0,0,1,0,0,0,
0,0,0,0,1,0,0,
0,0,0,0,0,1,0,
0,0,0,0,0,0,1,
1,0,0,0,0,0,0]
>;
a:=x;b:=y;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 64 [e.divs: 1.2^6].
B1:=MatrixAlgebra(F,7)!\[
3,-1,-1,1,1,-1,-1,
-1,3,-1,-1,1,1,-1,
-1,-1,3,-1,-1,1,1,
1,-1,-1,3,-1,-1,1,
1,1,-1,-1,3,-1,-1,
-1,1,1,-1,-1,3,-1,
-1,-1,1,1,-1,-1,3];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,7)!1;


L1:=MatrixAlgebra(Z,7)!B1;
LA1:=LatticeWithGram(L1);
AU1:=AutomorphismGroup(LA1);
// AU1 = <-x,y> = <x,-y> = <-x,-y> = 2 x S6(2).