ID Name Matrix 0 hkl [1 0 0; 0 1 0; 0 0 1] 1 0kl [0 0 0; 0 1 0; 0 0 1] 2 h0l [1 0 0; 0 0 0; 0 0 1] 3 hk0 [1 0 0; 0 1 0; 0 0 0] 4 h00 [1 0 0; 0 0 0; 0 0 0] 5 0k0 [0 0 0; 0 1 0; 0 0 0] 6 00l [0 0 0; 0 0 0; 0 0 1] 7 hhl [1 0 0; 1 0 0; 0 0 1] 8 h-hl [1 0 0; -1 0 0; 0 0 1] 9 hh0 [1 0 0; 1 0 0; 0 0 0] 10 hhh [1 0 0; 1 0 0; 1 0 0] ID Name Vector Multiplier 0 h == 2n [1 0 0] 2 1 k == 2n [0 1 0] 2 2 l == 2n [0 0 1] 2 3 h+k == 2n [1 1 0] 2 4 k+l == 2n [0 1 1] 2 5 h+l == 2n [1 0 1] 2 6 h+k+l == 2n [1 1 1] 2 7 2h+l == 4n [2 0 1] 4 8 l == 4n [0 0 1] 4 9 h+k == 4n [1 1 0] 4 10 k+l == 4n [0 1 1] 4 11 h+l == 4n [1 0 1] 4 12 -h+k+l == 3n [-1 1 1] 3 13 h-k+l == 3n [1 -1 1] 3 14 l == 3n [0 0 1] 3 15 l == 6n [0 0 1] 6 16 -h+l == 3n [-1 0 1] 3 17 h == 4n [1 0 0] 4 18 k == 4n [0 1 0] 4 19 h+l == 3n [1 0 1] 3 TRICLINIC 1, 2 0 1 -1 - P1 P-1 MonoclinicB 3, 3 Unique b axis 0 2 5 2 m 2/m - - - P121 P1m1 P12/m1 - - 1 P12_11 - P12_1/m1 - 0 - - P1a1 P12/a1 - 0 1 - - P12_1/a1 - 2 - - P1c1 P12/c1 - 2 1 - - P12_1/c1 - 5 - - P1n1 P12/n1 - 5 1 - - P12_1/n1 3 0 1 C121 C1m1 C12/m1 3 0, 2 1 - C1c1 C12/c1 4 2 1 A121 A1m1 A12/m1 4 0, 2 1 - A1n1 A12/n1 6 5 1 I121 I1m1 I12/m1 6 0, 2 1 - I1a1 I12/a1 MonoclinicC 3, 3 Unique c axis 0 3 6 2 m 2/m - - - P112 P11m P112/m - - 2 P112_1 - P112_1/m - 0 - - P11a P112/a - 0 2 - - P112_1/a - 1 - - P11b P112/b - 1 2 - - P112/n - 3 - - P11n P112_1/n - 3 2 - - P112/m 5 0 2 B112 B11m B112/n 5 0, 1 2 - B11n B112/m 4 1 2 A112 A11m A112/m 4 0, 1 2 - A11a A112/a 6 3 2 I112 I11m I112/m 6 0, 1 2 - I11b I112/b MonoclinicA 3, 3 Unique a axis 0 3 6 2 m 2/m - - - P211 Pm11 P2/m11 - - 0 P2_111 - P2_1/m11 - 1 - - Pb11 P2/b11 - 1 0 - - P2_1/b11 - 2 - - Pc11 P2/c11 - 2 0 - - P2_1/c11 - 4 - - Pn11 P2/n11 - 4 0 - - P2_1/n11 3 1 0 C211 Cm11 C2/m11 3 1, 2 0 - Cn11 C2/n11 5 2 0 B211 Bm11 B2/m11 5 1, 2 0 - Bb11 B2/b11 6 4 0 I211 Im11 I2/m11 6 1, 2 0 - Ic11 I2/c11 ORTHORHOMBIC 7, 3 0 1 2 3 4 5 6 222 mm2, m2m, 2mm mmm - - - - - - - P222 Pmm2, Pm2m, P2mm Pmmm - - - - - - 2 P222_1 - - - - - - - 1 - P22_12 - - - - - - - 1 2 P22_12_1 - - - - - - 0 - - P2_122 - - - - - - 0 - 2 P2_122_1 - - - - - - 0 1 - P2_12_12 - - - - - - 0 1 2 P2_12_12_1 - - - - - 0 0 - - - Pm2a, P2_1ma Pmma - - - 1 - 1 - - Pm2_1b, P2mb Pmmb - - - 3 0 1 - - Pm2_1n, P2_1mn Pmmn - - 0 - 0 - - - Pma2, P2_1am Pmam - - 0 0 0 - - - P2aa Pmaa - - 0 1 0 1 - - P2_1ab Pmab - - 0 3 0 1 - - P2an Pman - - 2 - - - 2 - Pmc2_1, P2cm Pmcm - - 2 0 0 - 2 - P2_1ca Pmca - - 2 1 - 1 2 - P2cb Pmcb - - 2 3 0 1 2 - P2_1cn Pmcn - - 5 - 0 - 2 - Pmn2_1, P2_1nm Pmnm - - 5 0 0 - 2 - P2na Pmna - - 5 1 0 1 2 - P2_1nb Pmnb - - 5 3 0 1 2 - P2nn Pmnn - 1 - - - 1 - - Pbm2, Pb2_1m Pbmm - 1 - 0 0 1 - - Pb2_1a Pbma - 1 - 1 - 1 - - Pb2b Pbmb - 1 - 3 0 1 - - Pb2n Pbmn - 1 0 - 0 1 - - Pba2 Pbam - 1 0 0 0 1 - - - Pbaa - 1 0 1 0 1 - - - Pbab - 1 0 3 0 1 - - - Pban - 1 2 - - 1 2 - Pbc2_1 Pbcm - 1 2 0 0 1 2 - - Pbca - 1 2 1 - 1 2 - - Pbcb - 1 2 3 0 1 2 - - Pbcn - 1 5 - 0 1 2 - Pbn2_1 Pbnm - 1 5 0 0 1 2 - - Pbna - 1 5 1 0 1 2 - - Pbnb - 1 5 3 0 1 2 - - Pbnn - 2 - - - - 2 - Pcm2_1, Pc2m Pcmm - 2 - 0 0 - 2 - Pc2a Pcma - 2 - 1 - 1 2 - Pc2_1b Pcmb - 2 - 3 0 1 2 - Pc2_1n Pcmn - 2 0 - 0 - 2 - Pca2_1 Pcam - 2 0 0 0 - 2 - - Pcaa - 2 0 1 0 1 2 - - Pcab - 2 0 3 0 1 2 - - Pcan - 2 2 - - - 2 - Pcc2 Pccm - 2 2 0 0 - 2 - - Pcca - 2 2 1 - 1 2 - - Pccb - 2 2 3 0 1 2 - - Pccn - 2 5 - 0 - 2 - Pcn2 Pcnm - 2 5 0 0 - 2 - - Pcna - 2 5 1 0 1 2 - - Pcnb - 2 5 3 0 1 2 - - Pcnn - 4 - - - 1 2 - Pnm2_1, Pn2_1m Pnmm - 4 - 0 0 1 2 - Pn2_1a Pnma - 4 - 1 - 1 2 - Pn2b Pnmb - 4 - 3 0 1 2 - Pn2n Pnmn - 4 0 - 0 1 2 - Pna2_1 Pnam - 4 0 0 0 1 2 - - Pnaa - 4 0 1 0 1 2 - - Pnab - 4 0 3 0 1 2 - - Pnan - 4 2 - - 1 2 - Pnc2 Pncm - 4 2 0 0 1 2 - - Pnca - 4 2 1 - 1 2 - - Pncb - 4 2 3 0 1 2 - - Pncn - 4 5 - 0 1 2 - Pnn2 Pnnm - 4 5 0 0 1 2 - - Pnna - 4 5 1 0 1 2 - - Pnnb - 4 5 3 0 1 2 - - Pnnn 3 1 0 3 0 1 - C222 Cmm2, Cm2m, C2mm Cmmm 3 1 0 3 0 1 2 C222_1 - - 3 1 0 0, 1 0 1 2 - Cm2a, C2mb Cmma 3 1 0, 2 3 0 1 2 - Cmc2_1, C2cm Cmmb 3 1 0, 2 0, 1 0 1 2 - C2cb Cmcm 3 1, 2 0 3 0 1 2 - Ccm2_1, Cc2m Cmca 3 1, 2 0 0, 1 0 1 2 - Cc2a Ccmm 3 1, 2 0, 2 3 0 1 2 - Ccc2 Ccmb 3 1, 2 0, 2 0, 1 0 1 2 - - Cccm 5 2 5 0 0 - 2 B222 Bmm2, Bm2m, B2mm Ccca, Cccb 5 2 5 0 0 1 2 B22_12 - Bmmm 5 2 5 0, 1 0 1 2 - Bm2_1b, B2mb Bmmb 5 2 0, 2 0 0 - 2 - Bma2, B2cm Bmam, Bmcm 5 2 0, 2 0, 1 0 1 2 - B2cb Bmab 5 1, 2 5 0 0 1 2 - Bbm2, Bb2_1m Bbmm 5 1, 2 5 0, 1 0 1 2 - Bb2b Bbmb 5 1, 2 0, 2 0 0 1 2 - Bba2 Bbcm 5 1, 2 0, 2 0, 1 0 1 2 - - Bbab, Bbcb 5 4 2 1 - 1 2 A222 Amm2, Am2m, A2mm Ammm 4 4 2 1 0 1 2 A2_122 - - 4 4 2 0, 1 0 1 2 - Am2a, A2_1ma Amma 4 4 0, 2 1 0 1 2 - Ama2, A2_1am Amam 4 4 0, 2 0, 1 0 1 2 - A2aa Amaa 4 1, 2 2 1 - 1 2 - Abm2, Ac2m Abmm, Acmm 4 1, 2 2 0, 1 0 1 2 - Ac2a Abma 4 1, 2 0, 2 1 0 1 2 - Aba2 Acam 4 1, 2 0, 2 0, 1 0 1 2 - - Abaa, Acaa 6 4 5 3 0 1 2 [I222, I2_12_12_1] Imm2, Im2m, I2mm Immm 6 4 5 0, 1 0 1 2 - Im2a, I2mb Imma, Immb 6 4 0, 2 3 0 1 2 - Ima2, I2cm Imam, Imcm 6 4 0, 2 0, 1 0 1 2 - I2cb Imcb 6 1, 2 5 3 0 1 2 - Ibm2, Ic2m Ibmm, Icmm 6 1, 2 5 0, 1 0 1 2 - Ic2a Icma 6 1, 2 0, 2 3 0 1 2 - Iba2 Ibam 6 1, 2 0, 2 0, 1 0 1 2 - - Ibca 3, 5, 4 4 5 3 0 1 2 F222 Fmm2, Fm2m, F2mm Fmmm 3, 5, 4 1, 2 11, 0, 2 9, 0, 1 17 18 8 - F2dd - 3, 5, 4 10, 1, 2 0, 2 9, 0, 1 17 18 8 - Fd2d - 3, 5, 4 10, 1, 2 11, 0, 2 0, 1 17 18 8 - Fdd2 - 3, 5, 4 10, 1, 2 11, 0, 2 9, 0, 1 17 18 8 - - Fddd Tetragonal 7, 7 0 3 1 7 6 5 9 4 -4 4/m 422 4mm -42m, -4m2 4/mmm - - - - - - - P4 P-4 P4/m P422 P4mm P-42m, P-4m2 P4/mmm - - - - - 1 - - - - P42_12 - P-42_1m - - - - - 2 - - P4_2 - P4_2/m P4_222 - - - - - - - 2 1 - - - - P4_22_12 - - - - - - - 8 - - {P4_1, P4_3} - - {P4_122, P4_322} - - - - - - - 8 1 - - - - {P4_12_12, P4_32_12} - - - - - - 2 2 - - - - - - P4_2mc P-42c P4_2/mmc - - - 2 2 1 - - - - - - P-42_1c - - - 1 - - 1 - - - - - P4bm P-4b2 P4/mbm - - 1 2 2 1 - - - - - P4_2bc - P4_2/mbc - - 2 - 2 - - - - - - P4_2cm P-4c2 P4_2/mcm - - 2 2 2 - - - - - - P4cc - P4/mcc - - 4 - 2 1 - - - - - P4_2nm P-4n2 P4_2/mnm - - 4 2 2 1 - - - - - P4nc - P4/mnc - 3 - - - 1 - - - P4/n - - - P4/nmm - 3 - - 2 1 - - - P4_2/n - - - - - 3 - 2 2 1 - - - - - - - P4_2/nmc - 3 1 - - 1 - - - - - - - P4/nbm - 3 1 2 2 1 - - - - - - - P4_2/nbc - 3 2 - 2 1 - - - - - - - P4_2/ncm - 3 2 2 2 1 - - - - - - - P4/ncc - 3 4 - 2 1 - - - - - - - P4_2/nnm - 3 4 2 2 1 - - - - - - - P4/nnc 6 3 4 2 2 1 - I4 I-4 I4/m I422 I4mm I-42m, I-4m2 I4/mmm 6 3 4 2 8 1 - I4_1 - - I4_122 - - - 6 3 4 7 8 1 0 - - - - I4_1md I-42d - 6 3 1, 2 2 2 1 - - - - - I4cm I-4c2 I4/mcm 6 3 1, 2 7 8 1 0 - - - - I4_1cd - - 6 0, 1 4 2 8 1 - - - I4_1/a - - - - 6 0, 1 4 7 8 1 0 - - - - - - I4_1/amd 6 0, 1 1, 2 7 8 1 0 - - - - - - I4_1/acd Trigonal 4, 8 0 8 7 6 3 -3 321, 32 3m1, 3m -3m1, -3 312 31m -31m - - - - P3 P-3 P321 P3m1 P-3ml P312 P31m P-31m - - - 14 {P3_1, P3_2} - {P3_121, P3_221} - - {P3_112, P3_212} - - - - 2 2 - - - - - - P31c P-31c - 2 - 2 - - - P3c1 P-3c1 - - - 12 19 14 14 R3 R-3 R32 R3m1 R-3m - - - 12 19, 2 14 15 - - - R3c R-3c - - - 13 16 14 14 R3 R-3 R32 R3m R-3m - - - 13 16, 2 14 15 - - - R3c R-3c - - - Trigonal(Rhom) 3, 5 0 7 10 3 -3 32 3m -3m - - - R3 R-3 R32 R3m R-3m - 2 0 - - - R3c R-3c Hexagonal 3, 7 8 7 6 6 -6 6/m 622 6mm -62m, -6m2 6/mmm - - - P6 P-6 P6/m P622 P6mm P-62m, P-6m2 P6/mmm - - 2 P6_3 - P6_3/m P6_322 - - - - - 14 {P6_2, P6_4} - - {P6_222, P6_422} - - - - - 15 {P6_1, P6_5} - - {P6_122, P6_522} - - - - 2 2 - - - - P6_3mc P-62c P6_3/mmc 2 - 2 - - - - P6_3cm P-6c2 P6_3/mcm 2 2 2 - - - - P6cc - P6/mcc Cubic 4, 5 0 1 7 6 23 m-3 432 -43m m-3m - - - - P23 Pm-3 P432 P-43m Pm-3m - - - 2 P2_13 - P4_232 - - - - - 8 - - {P4_132, P4_332} - - - - 2 2 - - - P-43n Pm-3n - 1 - 2 - Pa-3 - - - - 4 - 2 - Pn-3 - - Pn-3m - 4 2 2 - - - - Pn-3n 6 4 2 2 [I23, I2_13] Im-3 I432 I-43m Im-3m 6 4 2 8 - - I4_132 - - 6 4 7, 2 8 - - - I-43d - 6 2 2 2 - Ia-3 - - - 6 2 7, 2 8 - - - - Ia-3d 3, 5, 4 2 5 2 F23 Fm-3 F432 F-43m Fm-3m 3, 5, 4 2 5 8 - - F4_132 - - 3, 5, 4 2 0, 2 2 - - - F-43c Fm-3c 3, 5, 4 10, 1, 2 5 8 - Fd-3 - - Fd-3m 3, 5, 4 10, 1, 2 0, 2 8 - - - - Fd-3c