OpenWrite::noopen: Cannot open /stmp/mathematica/tmp_673271.txt. Content-type: text/html MGROUPS
Bilbao Crystallographic Server arrow SUBGROUPS Help

The magnetic space groups derived from the Fedorov space group: 075 > 075 (id = 6271)HHHHRead tr. matrix for the supergroup from /tmp/wp-supertrm6271.trnHHHH1.000000 0.000000 0.000000 0.000000 HHHH0.000000 1.000000 0.000000 0.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHinverted:HHHH1.000000 0.000000 0.000000 0.000000 HHHH0.000000 1.000000 0.000000 0.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHRead tr. matrix for the subgroup from /tmp/wp-subtrm6271.trnHHHH1.000000 0.000000 0.000000 0.000000 HHHH0.000000 1.000000 0.000000 0.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHinverted:HHHH1.000000 0.000000 0.000000 0.000000 HHHH0.000000 1.000000 0.000000 0.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHRead tr. matrix from /tmp/wp-trm6271.trnHHHH3.000000 0.000000 0.000000 1.000000 HHHH0.000000 3.000000 0.000000 1.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHTransformation matrix:HHHH3.000000 0.000000 0.000000 1.000000 HHHH0.000000 3.000000 0.000000 1.000000 HHHH0.000000 0.000000 1.000000 0.000000 HHHHdeterminant: 9.000000(det-ONE=8.000000)HHHHInverted of the transformation matrix:HHHH0.333333 0.000000 0.000000 -0.333333 HHHH0.000000 0.333333 0.000000 -0.333333 HHHH0.000000 0.000000 1.000000 0.000000 HHHHSubgroup generators: 1, general positions: 4HHHH(x,y,z)HHHH(-y,x,z)HHHH(y, -x,z)HHHH(-x, -y,z)HHHHSubgroup Wyckoff positions:HHHHList of Wyckoff positions:HHHHOrbit No:0HHHHMultiplicity:4HHHHLetter:dHHHH(x,y,z)HHHH(-y,x,z)HHHH(y, -x,z)HHHH(-x, -y,z)HHHHOrbit No:1HHHHMultiplicity:2HHHHLetter:cHHHH(0.000000, 0.500000,z)HHHH(0.500000, 0.000000,z)HHHHOrbit No:2HHHHMultiplicity:1HHHHLetter:bHHHH(0.500000, 0.500000,z)HHHHOrbit No:3HHHHMultiplicity:1HHHHLetter:aHHHH(0.000000, 0.000000,z)HHHHtr. matrix:(1.000000 + 3.000000x, 1.000000 + 3.000000y,z)HHHHdet = 9.000000HHHHSupergroup is transformed to unconventional setting.HHHHtake memory for 9 cells.HHHHtr. matrix:(1.000000 + 3.000000x, 1.000000 + 3.000000y,z)HHHHNumber of gen. pos. = 4, det = 9.000000. New number of general positions: 36HHHHchange the basis with the matrix (1.000000 + 3.000000x, 1.000000 + 3.000000y,z)HHHHAfter checking group we obtain 36 elementsHHHH000HHHHBefore expanding:HHHH(x,y,z)HHHH(0.333333 - x, 0.333333 - y,z)HHHH(0.333333 - y,x,z)HHHH(y, 0.333333 - x,z)HHHHAfter expanding general positions: number of cells: 1HHHHNumber of gen. pos. = 0, det = 9.000000. New number of general positions: 0HHHH(x,y,z)HHHH(x, 0.333333 +y,z)HHHH(x, 0.666667 +y,z)HHHH(0.333333 +x,y,z)HHHH(0.333333 +x, 0.333333 +y,z)HHHH(0.333333 +x, 0.666667 +y,z)HHHH(0.666667 +x,y,z)HHHH(0.666667 +x, 0.333333 +y,z)HHHH(0.666667 +x, 0.666667 +y,z)HHHH(0.333333 - x, 0.333333 - y,z)HHHH(0.333333 - x, 0.666667 - y,z)HHHH(0.333333 - x, -y,z)HHHH(0.666667 - x, 0.333333 - y,z)HHHH(0.666667 - x, 0.666667 - y,z)HHHH(0.666667 - x, -y,z)HHHH(-x, 0.333333 - y,z)HHHH(-x, 0.666667 - y,z)HHHH(-x, -y,z)HHHH(0.333333 - y,x,z)HHHH(0.333333 - y, 0.333333 +x,z)HHHH(0.333333 - y, 0.666667 +x,z)HHHH(0.666667 - y,x,z)HHHH(0.666667 - y, 0.333333 +x,z)HHHH(0.666667 - y, 0.666667 +x,z)HHHH(-y,x,z)HHHH(-y, 0.333333 +x,z)HHHH(-y, 0.666667 +x,z)HHHH(y, 0.333333 - x,z)HHHH(y, 0.666667 - x,z)HHHH(y, -x,z)HHHH(0.333333 +y, 0.333333 - x,z)HHHH(0.333333 +y, 0.666667 - x,z)HHHH(0.333333 +y, -x,z)HHHH(0.666667 +y, 0.333333 - x,z)HHHH(0.666667 +y, 0.666667 - x,z)HHHH(0.666667 +y, -x,z)HHHH000HHHHSupergroup is transformed (36 gen.pos.)->return the new group.HHHH(x,y,z)HHHH(x, 0.333333 +y,z)HHHH(x, 0.666667 +y,z)HHHH(0.333333 +x,y,z)HHHH(0.333333 +x, 0.333333 +y,z)HHHH(0.333333 +x, 0.666667 +y,z)HHHH(0.666667 +x,y,z)HHHH(0.666667 +x, 0.333333 +y,z)HHHH(0.666667 +x, 0.666667 +y,z)HHHH(0.333333 - x, 0.333333 - y,z)HHHH(0.333333 - x, 0.666667 - y,z)HHHH(0.333333 - x, -y,z)HHHH(0.666667 - x, 0.333333 - y,z)HHHH(0.666667 - x, 0.666667 - y,z)HHHH(0.666667 - x, -y,z)HHHH(-x, 0.333333 - y,z)HHHH(-x, 0.666667 - y,z)HHHH(-x, -y,z)HHHH(0.333333 - y,x,z)HHHH(0.333333 - y, 0.333333 +x,z)HHHH(0.333333 - y, 0.666667 +x,z)HHHH(0.666667 - y,x,z)HHHH(0.666667 - y, 0.333333 +x,z)HHHH(0.666667 - y, 0.666667 +x,z)HHHH(-y,x,z)HHHH(-y, 0.333333 +x,z)HHHH(-y, 0.666667 +x,z)HHHH(y, 0.333333 - x,z)HHHH(y, 0.666667 - x,z)HHHH(y, -x,z)HHHH(0.333333 +y, 0.333333 - x,z)HHHH(0.333333 +y, 0.666667 - x,z)HHHH(0.333333 +y, -x,z)HHHH(0.666667 +y, 0.333333 - x,z)HHHH(0.666667 +y, 0.666667 - x,z)HHHH(0.666667 +y, -x,z)HHHHSupergroup is transformed to the subgroup basis. there are 36 general positionsHHHHSupergroup generators: xxx, general positions:36HHHHSubgroup basis:HHHH(x,y,z)HHHH(x, 0.333333 +y,z)HHHH(x, 0.666667 +y,z)HHHH(0.333333 +x,y,z)HHHH(0.333333 +x, 0.333333 +y,z)HHHH(0.333333 +x, 0.666667 +y,z)HHHH(0.666667 +x,y,z)HHHH(0.666667 +x, 0.333333 +y,z)HHHH(0.666667 +x, 0.666667 +y,z)HHHH(0.333333 - x, 0.333333 - y,z)HHHH(0.333333 - x, 0.666667 - y,z)HHHH(0.333333 - x, -y,z)HHHH(0.666667 - x, 0.333333 - y,z)HHHH(0.666667 - x, 0.666667 - y,z)HHHH(0.666667 - x, -y,z)HHHH(-x, 0.333333 - y,z)HHHH(-x, 0.666667 - y,z)HHHH(-x, -y,z)HHHH(0.333333 - y,x,z)HHHH(0.333333 - y, 0.333333 +x,z)HHHH(0.333333 - y, 0.666667 +x,z)HHHH(0.666667 - y,x,z)HHHH(0.666667 - y, 0.333333 +x,z)HHHH(0.666667 - y, 0.666667 +x,z)HHHH(-y,x,z)HHHH(-y, 0.333333 +x,z)HHHH(-y, 0.666667 +x,z)HHHH(y, 0.333333 - x,z)HHHH(y, 0.666667 - x,z)HHHH(y, -x,z)HHHH(0.333333 +y, 0.333333 - x,z)HHHH(0.333333 +y, 0.666667 - x,z)HHHH(0.333333 +y, -x,z)HHHH(0.666667 +y, 0.333333 - x,z)HHHH(0.666667 +y, 0.666667 - x,z)HHHH(0.666667 +y, -x,z)HHHHnumber of cells: 1HHHH000HHHHSupergroup Wyckoff positions:HHHHOwn basis.HHHHList of Wyckoff positions:HHHHOrbit No:0HHHHMultiplicity:4HHHHLetter:dHHHH(x,y,z)HHHHOrbit No:1HHHHMultiplicity:2HHHHLetter:cHHHH(0.000000, 0.500000,z)HHHHOrbit No:2HHHHMultiplicity:1HHHHLetter:bHHHH(0.500000, 0.500000,z)HHHHOrbit No:3HHHHMultiplicity:1HHHHLetter:aHHHH(0.000000, 0.000000,z)HHHHSubgroup basis.HHHHList of Wyckoff positions:HHHHOrbit No:0HHHHMultiplicity:4HHHHLetter:dHHHH(1.000000 + 0.333333 (-1.000000 +x), 1.000000 + 0.333333 (-1.000000 +y),z)HHHHOrbit No:1HHHHMultiplicity:2HHHHLetter:cHHHH(0.666667, 0.833333,z)HHHHOrbit No:2HHHHMultiplicity:1HHHHLetter:bHHHH(0.833333, 0.833333,z)HHHHOrbit No:3HHHHMultiplicity:1HHHHLetter:aHHHH(0.666667, 0.666667,z)HHHHindex : 9HHHHSplitting of the general position of the supergroup...HHHHThe general position splits into 9 suborbits.HHHH&Splitting of all positions:HHHHGGGG4d: 4d 4d 4d 4d 4d 4d 4d 4d 4dHHHH2c: 4d 4d 4d 4d 2cHHHH1b: 4d 4d 1bHHHH1a: 4d 4d 1aHHHH&Splitting of Wyckoff position 4d: 4d 4d 4d 4d 4d 4d 4d 4d 4dHHHHOrbit 0:HHHH[0]TTTT(x,y,z)TTTT(0.333333 (-1.000000 +x), 0.333333 (-1.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000 - x, -1.000000 - y,z)TTTT(0.333333 (-2.000000 - x), 0.333333 (-2.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.000000 - y,x,z)TTTT(0.333333 (-2.000000 - y), 0.333333 (-1.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(y, -1.000000 - x,z)TTTT(0.333333 (-1.000000 +y), 0.333333 (-2.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHOrbit 1:HHHH[0]TTTT(x, -2.000000 +y,z)TTTT(0.333333 (-1.000000 +x), 0.333333 (-3.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000 - x, -2.000000 - y,z)TTTT(0.333333 (-2.000000 - x), -1.000000 - 0.333333y,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-2.000000 - y,x,z)TTTT(-1.000000 - 0.333333y, 0.333333 (-1.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-2.000000 +y, -1.000000 - x,z)TTTT(0.333333 (-3.000000 +y), 0.333333 (-2.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHOrbit 2:HHHH[0]TTTT(x, -1.000000 +y,z)TTTT(0.333333 (-1.000000 +x), 0.333333 (-2.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000 - x, -y,z)TTTT(0.333333 (-2.000000 - x), 0.333333 (-1.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-y,x,z)TTTT(0.333333 (-1.000000 - y), 0.333333 (-1.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-1.000000 +y, -1.000000 - x,z)TTTT(0.333333 (-2.000000 +y), 0.333333 (-2.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHOrbit 3:HHHH[0]TTTT(-2.000000 +x,y,z)TTTT(0.333333 (-3.000000 +x), 0.333333 (-1.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-2.000000 - x, -1.000000 - y,z)TTTT(-1.000000 - 0.333333x, 0.333333 (-2.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.000000 - y, -2.000000 +x,z)TTTT(0.333333 (-2.000000 - y), 0.333333 (-3.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(y, -2.000000 - x,z)TTTT(0.333333 (-1.000000 +y), -1.000000 - 0.333333x,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 4:HHHH[0]TTTT(-2.000000 +x, -2.000000 +y,z)TTTT(0.333333 (-3.000000 +x), 0.333333 (-3.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-2.000000 - x, -2.000000 - y,z)TTTT(-1.000000 - 0.333333x, -1.000000 - 0.333333y,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-2.000000 - y, -2.000000 +x,z)TTTT(-1.000000 - 0.333333y, 0.333333 (-3.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-2.000000 +y, -2.000000 - x,z)TTTT(0.333333 (-3.000000 +y), -1.000000 - 0.333333x,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 5:HHHH[0]TTTT(-2.000000 +x, -1.000000 +y,z)TTTT(0.333333 (-3.000000 +x), 0.333333 (-2.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-2.000000 - x, -y,z)TTTT(-1.000000 - 0.333333x, 0.333333 (-1.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-y, -2.000000 +x,z)TTTT(0.333333 (-1.000000 - y), 0.333333 (-3.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-1.000000 +y, -2.000000 - x,z)TTTT(0.333333 (-2.000000 +y), -1.000000 - 0.333333x,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 6:HHHH[0]TTTT(-1.000000 +x,y,z)TTTT(0.333333 (-2.000000 +x), 0.333333 (-1.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-x, -1.000000 - y,z)TTTT(0.333333 (-1.000000 - x), 0.333333 (-2.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.000000 - y, -1.000000 +x,z)TTTT(0.333333 (-2.000000 - y), 0.333333 (-2.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(y, -x,z)TTTT(0.333333 (-1.000000 +y), 0.333333 (-1.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHOrbit 7:HHHH[0]TTTT(-1.000000 +x, -2.000000 +y,z)TTTT(0.333333 (-2.000000 +x), 0.333333 (-3.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-x, -2.000000 - y,z)TTTT(0.333333 (-1.000000 - x), -1.000000 - 0.333333y,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-2.000000 - y, -1.000000 +x,z)TTTT(-1.000000 - 0.333333y, 0.333333 (-2.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-2.000000 +y, -x,z)TTTT(0.333333 (-3.000000 +y), 0.333333 (-1.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHOrbit 8:HHHH[0]TTTT(-1.000000 +x, -1.000000 +y,z)TTTT(0.333333 (-2.000000 +x), 0.333333 (-2.000000 +y),z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-x, -y,z)TTTT(0.333333 (-1.000000 - x), 0.333333 (-1.000000 - y),z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-y, -1.000000 +x,z)TTTT(0.333333 (-1.000000 - y), 0.333333 (-2.000000 +x),z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-1.000000 +y, -x,z)TTTT(0.333333 (-2.000000 +y), 0.333333 (-1.000000 - x),z)TTTT4dTTTT(y, -x,z)HHHHGGGG$Splitting of Wyckoff position 2c: 4d 4d 4d 4d 2cHHHHOrbit 0:HHHH[0]TTTT(0.000000, 0.500000,z)TTTT(-0.333333, -0.166667,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000, -1.500000,z)TTTT(-0.666667, -0.833333,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.500000, 0.000000,z)TTTT(-0.833333, -0.333333,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(0.500000, -1.000000,z)TTTT(-0.166667, -0.666667,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 1:HHHH[0]TTTT(0.000000, -1.500000,z)TTTT(-0.333333, -0.833333,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000, 0.500000,z)TTTT(-0.666667, -0.166667,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(0.500000, 0.000000,z)TTTT(-0.166667, -0.333333,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-1.500000, -1.000000,z)TTTT(-0.833333, -0.666667,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 2:HHHH[0]TTTT(0.000000, -0.500000,z)TTTT(-0.333333, -0.500000,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000, -0.500000,z)TTTT(-0.666667, -0.500000,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-0.500000, 0.000000,z)TTTT(-0.500000, -0.333333,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-0.500000, -1.000000,z)TTTT(-0.500000, -0.666667,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 3:HHHH[0]TTTT(-2.000000, 0.500000,z)TTTT(-1.000000, -0.166667,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-2.000000, -1.500000,z)TTTT(-1.000000, -0.833333,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.500000, -2.000000,z)TTTT(-0.833333, -1.000000,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(0.500000, -2.000000,z)TTTT(-0.166667, -1.000000,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 4:HHHH[0]TTTT(-2.000000, -0.500000,z)TTTT(-1.000000, -0.500000,z)TTTT2cTTTT(0.000000, 0.500000,z)HHHH[1]TTTT(-0.500000, -2.000000,z)TTTT(-0.500000, -1.000000,z)TTTT2cTTTT(0.500000, 0.000000,z)HHHHGGGG$Splitting of Wyckoff position 1b: 4d 4d 1bHHHHOrbit 0:HHHH[0]TTTT(0.500000, 0.500000,z)TTTT(-0.166667, -0.166667,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.500000, -1.500000,z)TTTT(-0.833333, -0.833333,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.500000, 0.500000,z)TTTT(-0.833333, -0.166667,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(0.500000, -1.500000,z)TTTT(-0.166667, -0.833333,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 1:HHHH[0]TTTT(0.500000, -0.500000,z)TTTT(-0.166667, -0.500000,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.500000, -0.500000,z)TTTT(-0.833333, -0.500000,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-0.500000, 0.500000,z)TTTT(-0.500000, -0.166667,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-0.500000, -1.500000,z)TTTT(-0.500000, -0.833333,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 2:HHHH[0]TTTT(-0.500000, -0.500000,z)TTTT(-0.500000, -0.500000,z)TTTT1bTTTT(0.500000, 0.500000,z)HHHHGGGG$Splitting of Wyckoff position 1a: 4d 4d 1aHHHHOrbit 0:HHHH[0]TTTT(0.000000, 0.000000,z)TTTT(-0.333333, -0.333333,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000, -1.000000,z)TTTT(-0.666667, -0.666667,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-1.000000, 0.000000,z)TTTT(-0.666667, -0.333333,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(0.000000, -1.000000,z)TTTT(-0.333333, -0.666667,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 1:HHHH[0]TTTT(0.000000, -2.000000,z)TTTT(-0.333333, -1.000000,z)TTTT4dTTTT(x,y,z)HHHH[1]TTTT(-1.000000, -2.000000,z)TTTT(-0.666667, -1.000000,z)TTTT4dTTTT(-x, -y,z)HHHH[2]TTTT(-2.000000, 0.000000,z)TTTT(-1.000000, -0.333333,z)TTTT4dTTTT(-y,x,z)HHHH[3]TTTT(-2.000000, -1.000000,z)TTTT(-1.000000, -0.666667,z)TTTT4dTTTT(y, -x,z)HHHHOrbit 2:HHHH[0]TTTT(-2.000000, -2.000000,z)TTTT(-1.000000, -1.000000,z)TTTT1aTTTT(0.000000, 0.000000,z)HHHHGGGG&{{4, d}, {2, c}, {1, b}, {1, a}}${{{4, d}, {4, d}, {4, d}, {4, d}, {4, d}, {4, d}, {4, d}, {4, d}, {4, d}}, {{4, d}, {4, d}, {4, d}, {4, d}, {2, c}}, {{4, d}, {4, d}, {1, b}}, {{4, d}, {4, d}, {1, a}}}${{{{x, y, z}, {-1 - x, -1 - y, z}, {-1 - y, x, z}, {y, -1 - x, z}}, {{x, -2 + y, z}, {-1 - x, -2 - y, z}, {-2 - y, x, z}, {-2 + y, -1 - x, z}}, {{x, -1 + y, z}, {-1 - x, -y, z}, {-y, x, z}, {-1 + y, -1 - x, z}}, {{-2 + x, y, z}, {-2 - x, -1 - y, z}, {-1 - y, -2 + x, z}, {y, -2 - x, z}}, {{-2 + x, -2 + y, z}, {-2 - x, -2 - y, z}, {-2 - y, -2 + x, z}, {-2 + y, -2 - x, z}}, {{-2 + x, -1 + y, z}, {-2 - x, -y, z}, {-y, -2 + x, z}, {-1 + y, -2 - x, z}}, {{-1 + x, y, z}, {-x, -1 - y, z}, {-1 - y, -1 + x, z}, {y, -x, z}}, {{-1 + x, -2 + y, z}, {-x, -2 - y, z}, {-2 - y, -1 + x, z}, {-2 + y, -x, z}}, {{-1 + x, -1 + y, z}, {-x, -y, z}, {-y, -1 + x, z}, {-1 + y, -x, z}}}, {{{0, 1/2, z}, {-1, -3/2, z}, {-3/2, 0, z}, {1/2, -1, z}}, {{0, -3/2, z}, {-1, 1/2, z}, {1/2, 0, z}, {-3/2, -1, z}}, {{0, -1/2, z}, {-1, -1/2, z}, {-1/2, 0, z}, {-1/2, -1, z}}, {{-2, 1/2, z}, {-2, -3/2, z}, {-3/2, -2, z}, {1/2, -2, z}}, {{-2, -1/2, z}, {-1/2, -2, z}}}, {{{1/2, 1/2, z}, {-3/2, -3/2, z}, {-3/2, 1/2, z}, {1/2, -3/2, z}}, {{1/2, -1/2, z}, {-3/2, -1/2, z}, {-1/2, 1/2, z}, {-1/2, -3/2, z}}, {{-1/2, -1/2, z}}}, {{{0, 0, z}, {-1, -1, z}, {-1, 0, z}, {0, -1, z}}, {{0, -2, z}, {-1, -2, z}, {-2, 0, z}, {-2, -1, z}}, {{-2, -2, z}}}}${{{{(-1 + x)/3, (-1 + y)/3, z}, {(-2 - x)/3, (-2 - y)/3, z}, {(-2 - y)/3, (-1 + x)/3, z}, {(-1 + y)/3, (-2 - x)/3, z}}, {{(-1 + x)/3, (-3 + y)/3, z}, {(-2 - x)/3, -1 - y/3, z}, {-1 - y/3, (-1 + x)/3, z}, {(-3 + y)/3, (-2 - x)/3, z}}, {{(-1 + x)/3, (-2 + y)/3, z}, {(-2 - x)/3, (-1 - y)/3, z}, {(-1 - y)/3, (-1 + x)/3, z}, {(-2 + y)/3, (-2 - x)/3, z}}, {{(-3 + x)/3, (-1 + y)/3, z}, {-1 - x/3, (-2 - y)/3, z}, {(-2 - y)/3, (-3 + x)/3, z}, {(-1 + y)/3, -1 - x/3, z}}, {{(-3 + x)/3, (-3 + y)/3, z}, {-1 - x/3, -1 - y/3, z}, {-1 - y/3, (-3 + x)/3, z}, {(-3 + y)/3, -1 - x/3, z}}, {{(-3 + x)/3, (-2 + y)/3, z}, {-1 - x/3, (-1 - y)/3, z}, {(-1 - y)/3, (-3 + x)/3, z}, {(-2 + y)/3, -1 - x/3, z}}, {{(-2 + x)/3, (-1 + y)/3, z}, {(-1 - x)/3, (-2 - y)/3, z}, {(-2 - y)/3, (-2 + x)/3, z}, {(-1 + y)/3, (-1 - x)/3, z}}, {{(-2 + x)/3, (-3 + y)/3, z}, {(-1 - x)/3, -1 - y/3, z}, {-1 - y/3, (-2 + x)/3, z}, {(-3 + y)/3, (-1 - x)/3, z}}, {{(-2 + x)/3, (-2 + y)/3, z}, {(-1 - x)/3, (-1 - y)/3, z}, {(-1 - y)/3, (-2 + x)/3, z}, {(-2 + y)/3, (-1 - x)/3, z}}}, {{{-1/3, -1/6, z}, {-2/3, -5/6, z}, {-5/6, -1/3, z}, {-1/6, -2/3, z}}, {{-1/3, -5/6, z}, {-2/3, -1/6, z}, {-1/6, -1/3, z}, {-5/6, -2/3, z}}, {{-1/3, -1/2, z}, {-2/3, -1/2, z}, {-1/2, -1/3, z}, {-1/2, -2/3, z}}, {{-1, -1/6, z}, {-1, -5/6, z}, {-5/6, -1, z}, {-1/6, -1, z}}, {{-1, -1/2, z}, {-1/2, -1, z}}}, {{{-1/6, -1/6, z}, {-5/6, -5/6, z}, {-5/6, -1/6, z}, {-1/6, -5/6, z}}, {{-1/6, -1/2, z}, {-5/6, -1/2, z}, {-1/2, -1/6, z}, {-1/2, -5/6, z}}, {{-1/2, -1/2, z}}}, {{{-1/3, -1/3, z}, {-2/3, -2/3, z}, {-2/3, -1/3, z}, {-1/3, -2/3, z}}, {{-1/3, -1, z}, {-2/3, -1, z}, {-1, -1/3, z}, {-1, -2/3, z}}, {{-1, -1, z}}}}${{{{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}}, {{{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{0, 1/2, z}, {1/2, 0, z}}}, {{{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{1/2, 1/2, z}}}, {{{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{x, y, z}, {-x, -y, z}, {-y, x, z}, {y, -x, z}}, {{0, 0, z}}}}&OK! (#)

Listed with respect to the BNS setting:


Listed with respect to the OG setting:



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