// ********************************************************************* // G = SL2(25) acting on Leech lattice G:=MatrixGroup<24, IntegerRing() | \[ 0, -1, -1, -1, -1, 2, 2, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 5, -6, -2, -1, 0, 2, -2, 1, 1, -2, 0, 0, 2, 2, -2, 2, 1, 1, -1, -1, 0, -2, 0, 0, -1, 3, 2, 1, -2, -1, 0, 0, 1, 0, 1, 0, -1, -1, 1, -1, 0, -1, 1, -1, 1, 2, 1, 1, 3, -5, -1, -2, 1, 2, -1, -1, 0, -2, -1, 0, 2, 2, -1, 2, 1, 1, 0, 0, 1, -1, 1, 2, 3, -2, -2, 1, -2, 0, -2, 2, 1, 0, 1, 0, 1, 1, -1, 0, 0, 0, 0, -1, 2, -2, 0, 1, 4, -3, 1, 0, -2, 1, -2, 0, 1, -2, 1, -1, 1, 0, 0, 1, 1, 1, 0, -2, 0, -1, 0, 0, 2, -2, -2, 2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 1, 1, 1, -3, -2, 1, 2, 1, 1, 0, -1, 0, -1, 1, 0, 1, -1, 0, 1, 1, -2, 1, 1, -1, 1, 2, 2, -2, 0, 1, -1, 2, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 2, 0, 0, 1, 1, -1, 0, 0, 0, 1, -1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, -3, -2, -2, 1, 1, -1, 1, 1, -1, -1, 0, 2, 1, -2, 1, 0, 0, 0, 0, 1, -2, 0, 1, 4, -3, -1, 0, -2, 1, -2, 2, 1, -1, 1, -1, 1, 1, -2, 0, 0, 1, 0, -1, 1, 1, 2, 2, 3, -5, -2, 0, 1, 2, 0, 0, 0, -1, -1, 1, 1, 2, -1, 1, 1, 1, -1, 0, -1, 0, 0, -1, -4, 4, 0, 2, 1, -2, 2, 0, -1, 2, 0, 1, -2, -1, 1, -2, -1, -1, 0, 1, 0, -1, 0, 0, -1, 1, 2, 0, 0, 1, 1, -1, 0, -1, -1, 1, -1, -1, 1, 0, 1, 0, 0, 0, -1, -1, -1, -2, -1, 1, -1, 2, 0, -2, 0, 1, -1, 1, 0, -1, -1, -1, 0, -1, -1, 0, -1, 0, -1, 0, 0, -2, -3, 2, -1, 2, 2, -2, 2, 0, -2, 1, -1, 0, -1, -1, 0, -2, -1, 0, -1, 1, 0, -2, -2, -2, -2, 4, 2, 0, -1, -1, 1, 0, 0, 1, 1, -1, -1, -2, 1, -1, -1, 0, 1, 0, 0, -1, 0, 1, -1, 3, 1, -1, -2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, 2, 0, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -2, 2, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, -2, 1, -1, 0, 0, 0, 0, 2, -1, 1, 1, 5, -6, 0, -1, -1, 2, -2, 0, 1, -3, 0, -1, 2, 1, -1, 2, 1, 1, 0, -2, -1, 1, 1, 1, -2, 1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, -2, 0, 0, -1, -4, 5, 0, 2, 0, -2, 2, 0, -1, 3, 0, 1, -2, -1, 1, -2, -1, -1, 0, 1 ], \[ 2, -2, 0, 0, 2, -1, 1, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 1, 0, -1, -2, 2, 0, 1, 0, -2, 0, 0, 0, 1, 1, 0, -1, 0, 1, -1, -1, -1, 0, 0, 1, -1, -1, 0, 0, 1, 1, 0, 1, 1, 2, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, -2, 1, 0, -1, -3, 2, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, -1, -1, 1, 0, 0, 2, 0, -1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -2, 0, 0, -1, 0, -1, 1, 0, -1, -1, 0, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -2, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, -1, 0, 0, 1, 0, 0, 2, -2, -1, 1, 2, 1, 1, 1, -2, 0, 0, 1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 1, 0, -2, 1, 0, 0, -3, 3, 2, -1, -1, 0, 0, -1, 1, 1, 1, 1, -1, 0, 2, 1, 0, -1, 1, 0, 0, 0, -1, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 0, 0, 0, -2, 1, 0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, 0, 1, -1, 0, 1, 1, 1, 2, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, -2, -2, 0, 2, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 2, 0, 1, -1, 0, 0, -3, 2, 2, 1, -1, 0, 0, -1, 1, 1, 1, -1, 2, 0, 0, 1, 1, 1, -1, -1, 0, 2, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, -1, 1, -1, 1, -2, 0, 2, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, -1, 0, 0, -1, -1, 1, -1, -1, 0, 1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, -3, 2, 0, -1, -3, 2, 0, 1, 1, -1, 1, -1, -1, 1, 0, 1, -2, 0, 1, -1, 0, -1, -1, 1, 0, 2, 1, 1, 1, -2, -1, -1, 1, 2, 0, -1, 0, -1, -1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, -4, -2, 0, 2, 1, 0, 0, -1, -1, -1, 0, 1, 1, -2, 1, 0, 2, -1, 0 ] /* order = 15600 = 2^4 * 3 * 5^2 * 13 */ >; // F is Gram matrix of Leech lattice F:=MatrixAlgebra(RationalField(), 24) ! [ RationalField() | 4, 0, 0, -2, -2, 2, -2, 0, -1, 2, -2, -1, -2, -1, 2, 1, -2, 2, -2, 2, 1, -1, -2, -1, 0, 4, -2, 1, 1, 1, 0, 2, -1, 0, 2, 1, 2, -2, -1, 0, -2, -2, -1, 2, -2, 2, 0, -2, 0, -2, 4, -2, 1, 1, 0, -2, 2, 1, -1, 1, -2, 1, 2, -1, 1, 1, 2, -1, 2, -1, 0, 1, -2, 1, -2, 4, 0, -1, 0, 0, -1, -2, 2, -1, 2, -1, -2, -1, 0, -2, 0, -1, -2, 2, 0, -1, -2, 1, 1, 0, 4, 0, 2, 0, 2, -1, 1, 2, 1, 1, -1, 0, 0, -2, 2, 0, -1, 0, 1, 1, 2, 1, 1, -1, 0, 4, -2, 0, 0, 2, -1, 0, -1, -2, 1, -1, -2, 1, -1, 2, 1, 1, -2, -2, -2, 0, 0, 0, 2, -2, 4, 1, 2, -1, 0, 2, 1, 2, -1, 1, 2, -1, 1, -1, -1, -1, 1, 2, 0, 2, -2, 0, 0, 0, 1, 4, 0, 1, 0, 0, 2, 0, -1, 0, 0, -1, -2, 2, -1, 0, 1, 0, -1, -1, 2, -1, 2, 0, 2, 0, 4, 1, -1, 2, 0, 2, 1, -1, 1, 0, 2, -1, 1, -1, 1, 1, 2, 0, 1, -2, -1, 2, -1, 1, 1, 4, -2, 0, -1, 0, 2, -1, -1, 1, -1, 1, 1, -1, -1, -1, -2, 2, -1, 2, 1, -1, 0, 0, -1, -2, 4, 1, 2, -1, -1, -1, 0, -2, 1, 0, -1, 2, 1, -1, -1, 1, 1, -1, 2, 0, 2, 0, 2, 0, 1, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, -2, 2, -2, 2, 1, -1, 1, 2, 0, -1, 2, 0, 4, 0, -2, -1, 0, -2, 0, 0, -2, 2, 1, 0, -1, -2, 1, -1, 1, -2, 2, 0, 2, 0, -1, 0, 0, 4, 0, 0, 2, 0, 1, -1, 1, -2, 1, 2, 2, -1, 2, -2, -1, 1, -1, -1, 1, 2, -1, 0, -2, 0, 4, 0, 0, 1, 0, 0, 2, -1, -1, 0, 1, 0, -1, -1, 0, -1, 1, 0, -1, -1, -1, 0, -1, 0, 0, 4, 0, 0, -1, 0, -1, -1, 0, 1, -2, -2, 1, 0, 0, -2, 2, 0, 1, -1, 0, 0, 0, 2, 0, 0, 4, 0, 1, -2, 1, -1, 1, 2, 2, -2, 1, -2, -2, 1, -1, -1, 0, 1, -2, 0, -2, 0, 1, 0, 0, 4, 0, 0, 2, -1, -1, 0, -2, -1, 2, 0, 2, -1, 1, -2, 2, -1, 1, 2, 0, 1, 0, -1, 1, 0, 4, -2, 0, 0, 1, 1, 2, 2, -1, -1, 0, 2, -1, 2, -1, 1, 0, 0, 0, -1, 0, 0, -2, 0, -2, 4, 0, 0, -1, -1, 1, -2, 2, -2, -1, 1, -1, -1, 1, 1, -1, 0, -2, 1, 2, -1, 1, 2, 0, 0, 4, -1, 0, 0, -1, 2, -1, 2, 0, 1, -1, 0, -1, -1, 2, 0, 2, -2, -1, -1, -1, -1, 0, 0, -1, 4, 0, -2, -2, 0, 0, 0, 1, -2, 1, 1, 1, -1, 1, 1, 1, 1, -1, 0, 1, -1, 1, -1, 0, 0, 4, 1, -1, -2, 1, -1, 1, -2, 2, 0, 1, -1, -1, 0, 0, 2, 0, 1, 2, 0, 1, -1, 0, -2, 1, 4 ]; // w7^2 = -7 in Endomorphism ring of G w7:=MatrixAlgebra(RationalField(), 24) ! [ RationalField() | 1, 0, 0, 4, -2, 2, 4, -2, 0, 4, 2, -2, 2, 0, 0, 4, 0, 2, 4, 2, 2, 0, 2, 0, 4, -3, 2, -2, 4, -6, 0, 4, 2, 2, 2, -2, -2, -4, -4, 0, -2, 0, 0, -2, 2, 4, -2, 0, 0, 4, -1, 2, -4, 4, -2, -2, 2, 0, 2, 0, -2, 4, 0, 0, 2, 0, 0, 0, -2, 0, 0, 4, 2, -2, 0, -3, 4, -4, 2, 0, 0, 0, 0, -2, 0, -4, -2, -2, 0, 0, -2, 0, 0, 0, 0, -2, 2, -2, -2, -2, 1, 2, 0, 0, -2, 0, 0, 2, -2, 2, 0, -4, 0, -2, -2, -2, -2, 2, 0, 2, 0, 0, 2, 0, -8, 5, 2, 0, 6, 2, 8, -4, -2, 2, -4, 4, -2, 0, 2, 0, -2, 0, 0, 2, -2, 0, 0, 0, 4, -4, 1, 0, -4, 2, -4, 0, 0, -2, 2, -2, 0, 0, 0, 2, 2, 2, 0, 0, -4, -2, 0, -2, -6, 8, 4, 3, -2, -2, 2, -2, 0, 2, 2, 2, -6, -2, 4, -2, 0, -2, 2, 0, -4, 2, 0, 2, -6, 8, 2, -2, -1, 0, 2, 0, 0, 4, 2, 0, -2, 0, 0, 0, -2, -2, 2, 4, -2, 2, 2, 2, -8, 6, 2, -2, 2, 1, 4, -2, 2, 2, 0, 4, -2, 0, 2, 0, 0, -2, 2, 2, 4, 0, 0, -2, 2, -2, -4, 4, 2, -2, 1, 2, -4, 0, -4, -2, 0, -2, -2, -4, 0, 2, -2, 2, 2, 2, 4, 4, 6, -8, -4, 2, 0, 4, -2, 1, -2, -2, -2, 0, 2, 2, -2, 0, 2, 4, -2, 2, 0, -8, -2, -6, 0, 4, 2, 6, -4, -6, 0, 2, -1, 0, 2, -4, -4, -4, 0, -4, -2, 0, 0, -2, -4, 0, -4, -2, -6, 10, 2, 0, -6, -6, -4, 4, 2, 5, 6, -4, -2, -4, 2, -2, -2, -4, 2, 0, 0, 4, 0, 6, -4, 4, 2, -4, 2, 4, 4, -2, 0, 2, -1, 4, 0, 2, 2, 2, 2, 0, 2, 4, 2, -2, 0, 2, 10, -10, 2, -2, -4, 4, -6, 0, 2, -4, 2, -1, 2, 2, 0, 4, 4, 4, 0, -4, -6, 4, -2, -2, -6, 6, -2, 0, 2, -4, 0, 0, -2, 4, 2, 0, -1, -2, 0, 0, -2, -2, 0, 2, -4, 2, 2, 2, -4, 2, -2, 2, 2, 0, 2, 0, 0, 2, 0, 4, 0, 1, 2, 0, 0, -2, -2, 0, 2, 0, -2, 0, 6, -4, -6, 0, 0, 0, -2, 4, -2, 0, 0, -4, 4, 0, -5, -2, -2, 2, -2, 2, 2, -2, 0, 2, -2, 4, 2, 2, -2, 2, 2, 0, 0, 2, 0, 2, -2, 0, 4, -1, 0, 0, 2, 0, -6, 6, 0, 2, -16, 16, 0, -2, 4, -4, 4, 0, 0, 8, 0, 4, -2, -2, 4, 0, -3, -6, 2, 4, 2, -4, 2, -4, 4, -8, -2, 6, 4, -2, 2, -2, -4, -4, -4, -2, -2, 0, -2, -2, 0, 3, -4, -2, -4, 4, 0, -2, -2, 4, -4, 2, 0, -6, -2, 2, -2, 4, 2, -2, -2, 0, -2, -4, -2, -2, -1, 2, -2, -2, -4, 2, 4, 2, 0, -2, -8, 0, -6, 4, 2, 2, 6, -2, 2, 0, 0, 2, 0, 0, 2, -1 ]; F:=MatrixRing(Integers(),24) ! F; w7:=MatrixRing(Rationals(),24) ! w7; al:=MatrixRing(Integers(),24) ! (1/2*(1+w7)); bl:=MatrixRing(Integers(),24) ! (1/2*(1-w7)); // Y gives the action of a Galois automorphism and is obtained by // X,Y:=IsIsometric([F,al*F],[F,bl*F]); Y:=MatrixAlgebra(IntegerRing(), 24) ! \[ 1, 1, 0, 1, 3, -3, -1, -1, 0, 1, -2, 0, 0, -1, 0, -1, 2, 1, -1, 1, 0, 1, 0, 0, 0, 0, 0, 1, -1, 2, 0, 0, -1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, -2, 0, -1, -1, 0, 2, -1, 1, 0, -1, 0, -1, -1, 0, 0, -2, -1, -1, 0, -1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, -1, 0, -1, -1, 0, 0, 0, 1, 0, 1, 1, 0, 1, -2, -2, 0, 2, 1, 1, 0, -1, 0, -1, 0, 1, 1, -2, 0, -1, 1, -1, 1, 1, 1, 0, 1, -1, 0, 0, -1, 1, 1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, -1, 0, -1, 0, -1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 1, 3, -4, -2, 0, 1, 1, 0, 1, 0, -1, -1, 0, 1, 1, -2, 0, 1, 1, -1, 0, 1, 1, 0, 0, 3, -3, -3, 0, 1, 0, -1, 1, -1, 0, 0, -1, 2, 1, -2, 0, -1, 1, -1, 0, -1, -1, 0, -1, -3, 4, 2, 1, -1, -2, 1, 0, 0, 1, 1, 0, -2, -1, 1, -1, -1, -1, 1, 0, 1, 0, 1, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2, -1, 0, 0, 2, 1, 0, -2, 0, 0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, 1, 1, 0, 3, -4, -2, 0, 1, 0, -1, 0, -1, -1, -1, -1, 1, 1, -2, 0, 0, 1, -1, 0, -1, 2, 0, 2, -2, 2, 1, -2, -1, 1, -1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, -1, 1, 1, 0, 0, 0, -1, -3, 3, 1, 0, 1, -1, 2, -1, 0, 1, -1, 1, -1, -1, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 2, -4, 0, 0, 1, 1, -1, -1, 0, -2, -1, 0, 1, 0, 0, 1, 1, 1, -1, -1, 1, -2, 0, -1, 0, 0, 1, 1, -1, -1, 0, 1, 0, 0, 0, -1, -1, -1, 0, -1, 0, 0, 0, -1, 1, 0, 0, 1, 3, -2, 0, 0, -1, 1, -1, 0, 0, -1, 1, -1, 1, 1, -1, 0, 0, 1, 0, 0, 0, 1, 1, -1, 0, -1, -1, 1, 2, -1, 1, -1, -1, 0, -1, 0, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, 0, -3, 4, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, -1, -1, 1, 0, -1, -1, 1, 1, -1, -1, 0, 0, -2, 3, 1, 0, -2, -2, -1, 1, 2, 1, 2, 0, -1, -1, 1, 0, 0, -2, 1, -1, 1, -1, -1, -1, 1, -1, 1, -1, -1, 0, -1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 1, 0, 0, -1 ]; // **************************************************************** // Construction of 72 dimensional lattice Git72 with Gram matrix Gram72 null:=MatrixRing(Integers(),24) ! 0; zwei:=MatrixRing(Integers(),24) ! 2; eins:=MatrixRing(Integers(),24) ! 1; FF:=DiagonalJoin(F,F); HH:=HorizontalJoin(bl,bl); NZ:=HorizontalJoin(null,zwei); FFF:=DiagonalJoin(F,FF); Lambda72:=LatticeWithGram(1/2*FFF); H:=HorizontalJoin(al,al); Hal:=HorizontalJoin(H,al); Hbet:=HorizontalJoin(null,HH); betH:=HorizontalJoin(HH,null); EA:=HorizontalJoin(eins,al); EE:=HorizontalJoin(eins,eins); EEa:=HorizontalJoin(EE,al); NNZ:=HorizontalJoin(null,NZ); L72:=VerticalJoin(EEa,Hbet); L72:=VerticalJoin(L72,NNZ); Git72:=sub; Determinant(Git72); Gram72:=L72*FFF*Transpose(L72); Gram72:=MatrixRing(Integers(),72) ! (1/2*MatrixRing(Rationals(),72) ! Gram72); // ********************************************************************* // The alpha as blockdiagonal matrix alpha72:=DiagonalJoin(al,al); alpha72:=DiagonalJoin(alpha72,al); // ******************************************************************** // GG72 generators of the group G on Leech + Leech + Leech = Lambda 72 GG:=Generators(G); GG72:=[]; for g in GG do h:=DiagonalJoin(g,g); h:=DiagonalJoin(h,g); Append(~GG72,h); end for; // ****************************************************************** // Construction of PSL2(7) as automorphism group of the Barnes lattice al2:=MatrixRing(Integers(),2) ! [0,1,-2,1]; bl2:=1-al2; null2:=MatrixRing(Integers(),2) ! 0; zwei2:=MatrixRing(Integers(),2) ! 2; eins2:=MatrixRing(Integers(),2) ! 1; F2:=MatrixRing(Integers(),2) ! [Trace(eins),Trace(al),Trace(bl),Trace(al*bl)]; FF2:=DiagonalJoin(F2,F2); FFF2:=DiagonalJoin(FF2,F2); alpha2:=DiagonalJoin(al2,al2); alpha2:=DiagonalJoin(alpha2,al2); Lambda6:=LatticeWithGram(FFF2); HH2:=HorizontalJoin(bl2,bl2); NZ2:=HorizontalJoin(null2,zwei2); H2:=HorizontalJoin(al2,al2); Hal2:=HorizontalJoin(H2,al2); Hbet2:=HorizontalJoin(null2,HH2); betH2:=HorizontalJoin(HH2,null2); EA2:=HorizontalJoin(eins2,al2); EE2:=HorizontalJoin(eins2,eins2); EEa2:=HorizontalJoin(EE2,al2); NNZ2:=HorizontalJoin(null2,NZ2); L6:=VerticalJoin(EEa2,Hbet2); L6:=VerticalJoin(L6,NNZ2); Git6:=sub< Lambda6 | L6 >; Determinant(Git6); Gram:=L6*FFF2*Transpose(L6); AL:=MatrixRing(Integers(),6) ! (MatrixRing(Rationals(),6) ! (L6*alpha2)*(MatrixRing(Rationals(),6) ! L6)^-1); AL eq alpha2; G6:=AutomorphismGroup([Gram,alpha2*Gram]); // Was needed to obtain the extra automorphism of Git72 x,y:=IsIsometric([Gram,alpha2*Gram],[Gram,(1-alpha2)*Gram]); a1:=MatrixRing(Rationals(),2) ! [1,0,1,-1]; aa1:=DiagonalJoin(a1,a1); aa1:=DiagonalJoin(aa1,a1); // ************************************************************* // Write the action of PSL2(7) on Git72 GG6:=Generators(G6); HH72:=[]; for g in GG6 do for i in [1..3] do a:=g[2*i-1][1]*eins+g[2*i-1][2]*al; b:=g[2*i-1][3]*eins+g[2*i-1][4]*al; c:=g[2*i-1][5]*eins+g[2*i-1][6]*al; hi:=HorizontalJoin(a,b); hi:=HorizontalJoin(hi,c); if i eq 1 then h:=hi; else h:=VerticalJoin(h,hi); end if; end for; Append(~HH72,h); end for; // ******************************************************************** GH72:=GG72 cat HH72; // GG72 generate SL2(25), HH72 the PSL2(7) for h in GH72 do h*Gram72*Transpose(h) eq Gram72; alpha72*h eq h*alpha72; end for; //******************************************************************** // Construct extra automorphism Z1:=HorizontalJoin(Y,-Y); Z1:=HorizontalJoin(Z1,al*Y); Z2:=HorizontalJoin(0*Y,-bl*Y); Z2:=HorizontalJoin(Z2,Y); Z3:=HorizontalJoin(0*Y,Y); Z3:=HorizontalJoin(-al*Y,Z3); Z:=VerticalJoin(Z1,Z2); Z:=VerticalJoin(Z,Z3); Z*Gram72*Transpose(Z) eq Gram72; alpha72*Z eq Z*(1-alpha72); Append(~GH72,Z); //******************************************************************** // G72 = (PSL2(7) x SL2(25)):2 , probably Aut(Git32) G72:=sub; //********************************************************************** // Check that G72 is almost everywhere irreducible g72mod2:=ChangeRing(G72,GF(2)); m2:=GModule(g72mod2); CompositionFactors(m2); g72mod3:=ChangeRing(G72,GF(3)); m3:=GModule(g72mod3); CompositionFactors(m3); g72mod5:=ChangeRing(G72,GF(5)); m5:=GModule(g72mod5); CompositionFactors(m5); g72mod7:=ChangeRing(G72,GF(7)); m7:=GModule(g72mod7); CompositionFactors(m7); g72mod13:=ChangeRing(G72,GF(13)); m13:=GModule(g72mod13); CompositionFactors(m13);