# F:=Rationals(); local result, l; result:= rec(); result.comment:= "2\"A5 as 8 x 8 matrices over Z.\n\ Irreducible over Q, becomes 4 + 4 over C.\n\ Schur Index 2 (for the irreducible constituents).\n\ \n\ SEED:\n\ Undefined.\n\ v [= e1] has 2 x 20 = 40 images under G; has 20 images under G.\n\ BASIS:\n\ 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.\n\ \n\ Possible matrix entries for any element of the group:\n\ The 3 elements of {-1,0,1} only.\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 32 (50% exactly).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 32 50\n\ 1 16 25\n\ -1 16 25\n\ nonzero 32 50\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 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] ], l -> List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.symmetricforms, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.antisymmetricforms, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.antisymmetricforms, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.antisymmetricforms, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.centralizeralgebra, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.centralizeralgebra, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.centralizeralgebra, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 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]; Add( result.centralizeralgebra, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); return result;