/*
2"A10 as 16 x 16 matrices over Z.
Absolutely irreducible representation.
Schur index 1.

SEED:
Nonzero v fixed by <y^3,x*y^3*x*y^-2*x> = L2(8).
(Stabiliser of v is L2(8):3, and of <v> is 2 x L2(8):3.)
v has 2 x 1200 = 2400 images under G; <v> has 1200 images under G.
BASIS:
All in v^G.

Possible matrix entries are in {-2,-1,0,1,2}.

Average number of nonzero entries for any element of the group:
109 + 11/25  (109.44; 42.75% exactly).

  Entry    Av/Mat                 %Av/Mat
      0   146.56  [146 + 14/25]    57.25  [57 + 1/4]
nonzero   109.44  [109 + 11/25]    42.75  [42 + 3/4]
     ±1   107.52  [107 + 13/25]    42     [42      ]
     ±2     1.92  [  1 + 23/25]     0.75  [   + 3/4]
      1    53.76  [ 53 + 19/25]    21     [21      ]
     -1    53.76  [ 53 + 19/25]    21     [21      ]
      2     0.96  [      24/25]     0.375 [   + 3/8]
     -2     0.96  [      24/25]     0.375 [   + 3/8]


p-MODULAR REDUCTIONS:
p = 5: 8a.8b (uniserial, both factors absolutely irreducible).
       Module labelling should be the same as elsewhere in the web-ATLAS.
other: absolutely irreducible.


LATTICE DETAILS:
Aut grp: 2"A10.
Minimum: 6.
Kiss no: 2400.
Determinant: 390625.
Elmnty divs: 1^8.5^8.

Theta series:
1 + 2400*q^6 + 10800*q^8 + 60480*q^10 + 223200*q^12 + 626400*q^14 +
1609200*q^16 + 3681600*q^18 + 7801920*q^20 + 14904000*q^22 + 27729600*q^24 +
48168000*q^26 + 81410400*q^28 + 132030240*q^30 + O(q^31).

Minimum vectors are all in v^G.
*/


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

G<x,y>:=MatrixGroup<16,F|\[
0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1]
,\[
0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,
-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,
0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,-1,
0,0,0,0,0,0,0,1,0,0,0,0,0,1,-1,0,
0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,
0,1,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,
1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,-1,-1,0,-1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,
1,0,1,0,0,0,0,1,-1,-1,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1]
>;
A:=x;B:=y;


// Forms: B1 (Symmetric).
// B1 (Symmetric form): Determinant 390625 [e.divs: 1^8.5^8].
B1:=MatrixAlgebra(F,16)!\[
6,-3,-3,2,3,-1,1,0,1,2,1,0,-3,3,0,0,
-3,6,1,-3,-2,3,-1,1,-3,1,-1,1,0,-3,0,0,
-3,1,6,-3,-3,0,2,-3,1,-1,0,-3,1,0,-3,2,
2,-3,-3,6,3,-3,1,2,0,1,3,0,-1,1,1,-3,
3,-2,-3,3,6,-3,-1,3,0,3,2,0,-1,0,0,0,
-1,3,0,-3,-3,6,-2,-1,-3,0,-2,2,1,-1,0,0,
1,-1,2,1,-1,-2,6,-3,2,0,3,-3,-3,3,-2,-1,
0,1,-3,2,3,-1,-3,6,-2,2,0,3,0,-3,3,-2,
1,-3,1,0,0,-3,2,-2,6,-3,0,-3,0,3,0,0,
2,1,-1,1,3,0,0,2,-3,6,3,0,-3,0,0,0,
1,-1,0,3,2,-2,3,0,0,3,6,-3,-3,2,-1,-2,
0,1,-3,0,0,2,-3,3,-3,0,-3,6,1,-3,3,-1,
-3,0,1,-1,-1,1,-3,0,0,-3,-3,1,6,-3,-1,1,
3,-3,0,1,0,-1,3,-3,3,0,2,-3,-3,6,0,-1,
0,0,-3,1,0,0,-2,3,0,0,-1,3,-1,0,6,-3,
0,0,2,-3,0,0,-1,-2,0,0,-2,-1,1,-1,-3,6];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,16)!1;


L1:=MatrixAlgebra(Z,16)!B1;
LA1:=LatticeWithGram(L1);
// AU1:=AutomorphismGroup(LA1);
// AU1 = G = 2"A10.