/* 2"A5 as 8 x 8 matrices over Z. Irreducible over Q, becomes 4 + 4 over C. Schur Index 2 (for the irreducible constituents). SEED: Undefined. v [= e1] has 2 x 20 = 40 images under G; has 20 images under G. BASIS: v, v.x, v.y, v.x*y, v.y^2, v.x*y^2, v.x^3*y*x*y^2*x, v.y*x*y^2*x. Possible matrix entries for any element of the group: The 3 elements of {-1,0,1} only. Average number of nonzero entries for any element of the group: 32 (50% exactly). Entry Av/Mat %Av/Mat 0 32 50 1 16 25 -1 16 25 nonzero 32 50 */ F:=Rationals(); G:=MatrixGroup<8,F|\[ 0,1,0,0,0,0,0,0, -1,0,0,0,0,0,0,0, -1,0,0,1,-1,1,-1,-1, 0,-1,-1,0,-1,-1,1,-1, 1,0,-1,1,-1,-1,0,-1, 0,1,-1,-1,1,-1,1,0, -1,0,-1,-1,0,-1,1,-1, 0,-1,1,-1,1,0,1,1] ,\[ 0,0,1,0,0,0,0,0, 0,0,0,1,0,0,0,0, 0,0,0,0,1,0,0,0, 0,0,0,0,0,1,0,0, 1,0,0,0,0,0,0,0, 0,1,0,0,0,0,0,0, 0,0,0,0,0,0,1,0, 0,0,0,0,0,0,0,1] >; A:=x;B:=y; // Forms: B1 (Symmetric); B2,B3,B4 (Antisymmetric). // The antisymmetric forms span a 3-space. // a*B2+b*B3+c*B4 has determinant (a^2-3*a*b+3*b^2+3*c^2)^4. // B1 (Symmetric form): Determinant 1. B1:=MatrixAlgebra(F,8)!\[ 2,0,0,-1,0,1,0,-1, 0,2,1,0,-1,0,1,0, 0,1,2,0,0,-1,0,-1, -1,0,0,2,1,0,1,0, 0,-1,0,1,2,0,0,-1, 1,0,-1,0,0,2,1,0, 0,1,0,1,0,1,2,0, -1,0,-1,0,-1,0,0,2]; // B2 (Antisymmetric form): Determinant 1. B2:=MatrixAlgebra(F,8)!\[ 0,1,0,0,0,0,0,0, -1,0,0,1,0,-1,0,1, 0,0,0,1,0,0,0,0, 0,-1,-1,0,0,1,0,1, 0,0,0,0,0,1,0,0, 0,1,0,-1,-1,0,0,1, 0,0,0,0,0,0,0,1, 0,-1,0,-1,0,-1,-1,0]; // B3 (Antisymmetric form): Determinant 81. B3:=MatrixAlgebra(F,8)!\[ 0,0,1,0,-1,0,1,0, 0,0,0,-1,0,1,0,-1, -1,0,0,0,1,0,1,0, 0,1,0,0,0,-1,0,-1, 1,0,-1,0,0,0,1,0, 0,-1,0,1,0,0,0,-1, -1,0,-1,0,-1,0,0,0, 0,1,0,1,0,1,0,0]; // B4 (Antisymmetric form): Determinant 81. B4:=MatrixAlgebra(F,8)!\[ 0,0,0,1,0,-1,0,1, 0,0,1,0,-1,0,1,0, 0,-1,0,0,0,1,0,1, -1,0,0,0,1,0,1,0, 0,1,0,-1,0,0,0,1, 1,0,-1,0,0,0,1,0, 0,-1,0,-1,0,-1,0,0, -1,0,-1,0,-1,0,0,0]; BS1:=B2;BS2:=B2+B3;BS3:=B4; // a*BS1+b*BS1+c*BS3 has determinant (a^2-a*b+b^2+3*c^2)^4. // Centralising algebra: 4-dimensional. // C1,C2,C3,C4 correspond respectively to the quaternions 1,i,r3*j,r3*k. C1:=MatrixAlgebra(F,8)!\[ 1,0,0,0,0,0,0,0, 0,1,0,0,0,0,0,0, 0,0,1,0,0,0,0,0, 0,0,0,1,0,0,0,0, 0,0,0,0,1,0,0,0, 0,0,0,0,0,1,0,0, 0,0,0,0,0,0,1,0, 0,0,0,0,0,0,0,1]; C2:=MatrixAlgebra(F,8)!\[ 0,1,0,0,0,0,0,0, -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, 0,0,0,0,-1,0,0,0, 0,0,0,0,0,0,0,1, 0,0,0,0,0,0,-1,0]; C3:=MatrixAlgebra(F,8)!\[ 0,3,-2,0,2,0,-2,0, 3,0,0,2,0,-2,0,2, 2,0,0,3,-2,0,-2,0, 0,-2,3,0,0,2,0,2, -2,0,2,0,0,3,-2,0, 0,2,0,-2,3,0,0,2, 2,0,2,0,2,0,0,3, 0,-2,0,-2,0,-2,3,0]; C4:=MatrixAlgebra(F,8)!\[ 3,0,0,2,0,-2,0,2, 0,-3,2,0,-2,0,2,0, 0,-2,3,0,0,2,0,2, -2,0,0,-3,2,0,2,0, 0,2,0,-2,3,0,0,2, 2,0,-2,0,0,-3,2,0, 0,-2,0,-2,0,-2,3,0, -2,0,-2,0,-2,0,0,-3];