1gap> pg:=SymplecticSpace(5,2); 2W(5, 2) 3gap> pi:=Random(Planes(pg)); 4<a plane in W(5, 2)> 5gap> l:=Random(Lines(pi)); 6<a line in W(5, 2)> 7gap> p:=Random(Points(l)); 8<a point in W(5, 2)> 9gap> g:=CollineationGroup(pg); 10PGammaSp(6,2) 11gap> g1:=Stabilizer(g,p); 12<projective collineation group of size 23040 with 3 generators> 13gap> g2:=Stabilizer(g,l); 14<projective collineation group of size 4608 with 4 generators> 15gap> g3:=Stabilizer(g,pi); 16<projective collineation group of size 10752 with 3 generators> 17gap> cg:=CosetGeometry(g, [g1,g2,g3]); 18CosetGeometry( PGammaSp(6,2) ) 19gap> RandomFlag(cg); # Time of last command: 10745 ms 20<Flag of coset geometry < CosetGeometry( PGammaSp(6,2) , 21[ 22 Group( 23 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 24 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ], 25 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 26 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 27 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 28 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 29 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) 30 ],]),IdentityMapping( GF(2) )), 31 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 32 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 33 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 34 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 35 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 36 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 37 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) 38 ],]),IdentityMapping( GF(2) )), 39 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 40 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], 41 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 42 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 43 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], 44 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 45 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) 46 ],]),IdentityMapping( GF(2) )) ] ), 47 Group( 48 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 49 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 50 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 51 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 52 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 53 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 54 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) 55 ],]),IdentityMapping( GF(2) )), 56 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 57 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], 58 [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ], 59 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], 60 [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ], 61 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], 62 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 63 ],]),IdentityMapping( GF(2) )), 64 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 65 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 66 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 67 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ], 68 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ], 69 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 70 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 71 ],]),IdentityMapping( GF(2) )), 72 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 73 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 74 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], 75 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 76 [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ], 77 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 78 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) 79 ],]),IdentityMapping( GF(2) )) ] ), 80 Group( 81 [ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 82 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 83 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 84 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 85 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 86 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 87 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) 88 ],]),IdentityMapping( GF(2) )), 89 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 90 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 ], 91 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 92 [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], 93 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 94 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 95 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 96 ],]),IdentityMapping( GF(2) )), 97 ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 98 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 99 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], 100 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 101 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 102 [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 103 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) 104 ],]),IdentityMapping( GF(2) )) ] ) ] ) >> 105gap> Type(last); 106[ ] 107gap> ResidueOfFlag(last2); 108CosetGeometry( PGammaSp(6,2) ) 109gap> Rank(last); 1103 111gap> NrElementsOfIncidenceStructure(last2,1); 11263 113gap> flg:=RandomFlag(cg);; 114gap> can:=CanonicalResidueOfFlag(cg,flg); 115CosetGeometry( Group( ... ) ) 116gap> Type(flg); 117[ 1, 2 ] 118gap> Rank(can); 1191 120gap> res:=ResidueOfFlag(flg); 121CosetGeometry( Group( 122[ ProjElWithFrob(NewMatrix(IsCMatRep,GF(2,1),6,[ 123 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 124 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], 125 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 126 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 127 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], 128 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2) 129 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2, 130 1),6,[[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 131 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 132 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 133 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 134 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 135 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 136 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2, 137 1),6,[[ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 138 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 139 [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ], 140 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 141 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 142 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0 143 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2, 144 1),6,[[ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 145 [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], 146 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], 147 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ], 148 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], 149 [ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2) 150 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2, 151 1),6,[[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 152 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ], 153 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 154 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 155 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 156 [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) 157 ],]),IdentityMapping( GF(2) )), ProjElWithFrob(NewMatrix(IsCMatRep,GF(2, 158 1),6,[[ Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], 159 [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ], 160 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], 161 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 162 [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 163 [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) 164 ],]),IdentityMapping( GF(2) )) ] ) ) 165gap> IsIsomorphicIncidenceStructureWithNauty(res,can); 166#I Using NiceMonomorphism... 167#I Using NiceMonomorphism... 168true 169