International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 14.2, pp. 848-872
https://doi.org/10.1107/97809553602060000531

## Chapter 14.2. Symbols and properties of lattice complexes

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:  kochelke@mailer.uni-marburg.de

The reference symbols of lattice complexes are described and the terms degrees of freedom' of a lattice complex, limiting complex', comprehensive complex' and Weissenberg complex' are defined and illustrated by examples. Then the descriptive symbols of lattice complexes are introduced, their properties are described and their interpretation is demonstrated by numerous examples. Tables 14.2.3.1 and 14.2.3.2 give the explicit assignment of the Wyckoff positions of all plane groups and space groups, respectively, to Wyckoff sets and to lattice complexes. For each Wyckoff position, the reference symbol of the corresponding lattice complex is tabulated. In addition, a descriptive symbol is given that describes the arrangement of points in the corresponding point configurations. It refers directly to the coordinate description of the Wyckoff position.

### 14.2.1. Reference symbols and characteristic Wyckoff positions

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If a lattice complex can be generated in different space-group types, one of them stands out because its corresponding Wyckoff positions show the highest site symmetry. This is called the characteristic space-group type of the lattice complex. The space groups of all the other types in which the lattice complex may be generated are subgroups of the space groups of the characteristic type.

Different lattice complexes may have the same characteristic space-group type but in that case they differ in the oriented site symmetry of their Wyckoff positions within the space groups of that type.

The characteristic space-group type and the corresponding oriented site symmetry express the common symmetry properties of all point configurations of a lattice complex. Therefore, they can be used to identify each lattice complex. Within the reference symbols of lattice complexes, however, instead of the site symmetry the Wyckoff letter of one of the Wyckoff positions with that site symmetry is given, as was first done by Hermann (1935). This Wyckoff position is called the characteristic Wyckoff position of the lattice complex.

#### Examples

 (1) is the characteristic space-group type for the lattice complex of all cubic primitive point lattices. The Wyckoff positions with the highest possible site symmetry are la 000 and 1b , from which 1a has been chosen as the characteristic position. Thus, the lattice complex is designated . (2) is also characteristic for another lattice complex that corresponds to Wyckoff position 8g xxx .3m. Thus, the reference symbol for this lattice complex is . Each of its point configurations may be derived by replacing each point of a cubic primitive lattice by eight points arranged at the corners of a cube.

In Tables 14.2.3.1 and 14.2.3.2, the reference symbols denote the lattice complex of each Wyckoff position. The reference symbols of characteristic Wyckoff positions are marked by asterisks (e.g. 2e in ). If in a particular space group several Wyckoff positions belong to the same Wyckoff set (cf. Koch & Fischer, 1975), the reference symbol is given only once (e.g. Wyckoff positions 4l to 4o in ). To enable this, the usual sequence of Wyckoff positions had to be changed in a few cases (e.g. in ). For Wyckoff positions assigned to the same lattice complex but belonging to different Wyckoff sets, the reference symbol is repeated. In , for example, Wyckoff positions 4c and 4d are both assigned to the lattice complex . They do not belong, however, to the same Wyckoff set because the site-symmetry groups .. of 4c and .. of 4d are different.

 Table 14.2.3.1| top | pdf | Plane groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes
 Wyckoff positions of the same Wyckoff set can be recognized by their consecutive listing without repetition of the reference symbol. Characteristic Wyckoff sets are marked by asterisks.
 1 p1 1 a 1 p2 a P[xy] 2 p2 1 a 2 * P 1 b 1 c 1 d 2 e 1 * P2xy 3 pm 1 a .m. p2mm a P[y] 1 b 2 c 1 p2mm e P2x[y] 4 pg 2 a 1 p2mg c 5 cm 2 a .m. c2mm a C[y] 4 b 1 c2mm d C2x[y] 6 p2mm 1 a 2mm * P 1 b 1 c 1 d 2 e ..m * P2x 2 f 2 g .m. P2y 2 h 4 i 1 * P2x2y 7 p2mg 2 a 2.. p2mm a 2 b 2 c .m. * 4 d 1 * 8 p2gg 2 a 2.. c2mm a C 2 b 4 c 1 * .g. C2xy 9 c2mm 2 a 2mm * C 2 b 4 c 2.. p2mm a 4 d ..m * C2x 4 e .m. C2y 8 f 1 * C2x2y 10 p4 1 a 4.. p4mm a P 1 b 2 c 2.. p4mm a 4 d 1 * P4xy 11 p4mm 1 a 4mm * P 1 b 2 c 2mm. p4mm a 4 d .m. * P4x 4 e 4 f ..m * P4xx 8 g 1 * P4x2y 12 p4gm 2 a 4.. p4mm a C 2 b 2.mm p4mm a 4 c ..m * 8 d 1 * ..m C4xy 13 p3 1 a 3.. p6mm a P 1 b 1 c 3 d 1 * P3xy 14 p3m1 1 a 3m. p6mm a P 1 b 1 c 3 d .m. * 6 e 1 * 15 p31m 1 a 3.m p6mm a P 2 b 3.. p6mm b G 3 c ..m * P3x 6 d 1 * P3x2y 16 p6 1 a 6.. p6mm a P 2 b 3.. p6mm b G 3 c 2.. p6mm c N 6 d 1 * P6xy 17 p6mm 1 a 6mm * P 2 b 3m. * G 3 c 2mm * N 6 d ..m * P6x 6 e .m. * 12 f 1 * P6x2y
 Table 14.2.3.2| top | pdf | Space groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes
 Wyckoff positions of the same Wyckoff set can be recognized by their consecutive listing without repetition of the reference symbol. Characteristic Wyckoff sets are marked by asterisks.
 1 P1 1 a 1 P[xyz] 2 1 a P 1 b 1 c 1 d 1 e 1 f 1 g 1 h 2 i 1 * P2xyz 3 P2 1 a 2 P[y] 1 b 1 c 1 d 2 e 1 P2xz[y] 4 2 a 1 5 C2 2 a 2 C[y] 2 b 4 c 1 C2xz[y] 6 Pm 1 a m P[xz] 1 b 2 c 1 P2y[xz] 7 Pc 2 a 1 8 Cm 2 a m C[xz] 4 b 1 C2y[xz] 9 Cc 4 a 1 10 1 * P 1 b 1 c 1 d 1 e 1 f 1 g 1 h 2 i 2 * P2y 2 j 2 k 2 l 2 m m * P2xz 2 n 4 o 1 * P2xz2y 11 2 a 2 b 2 c 2 d 2 e m * 4 f 1 * 12 2 a * C 2 b 2 c 2 d 4 e 4 f 4 g 2 * C2y 4 h 4 i m * C2xz 8 j 1 * C2xz2y 13 2 a 2 b 2 c 2 d 2 e 2 * 2 f 4 g 1 * 14 2 a A 2 b 2 c 2 d 4 e 1 * c A2xyz 15 4 a 4 b 4 c 4 d 4 e 2 * 8 f 1 * 16 P222 1 a 222 Pmmm a P 1 b 1 c 1 d 1 e 1 f 1 g 1 h 2 i 2.. Pmmm i P2x 2 j 2 k 2 l 2 m .2. P2y 2 n 2 o 2 p 2 q ..2 P2z 2 r 2 s 2 t 4 u 1 * P2x2yz 17 2 a 2.. Pmma e 2 b 2 c .2. 2 d 4 e 1 * 18 2 a ..2 Pmmn a 2 b 4 c 1 * 19 4 a 1 * 20 4 a 2.. Cmcm c 4 b .2. 8 c 1 * 21 C222 2 a 222 Cmmm a C 2 b 2 c 2 d 4 e 2.. Cmmm g C2x 4 f 4 g .2. C2y 4 h 4 i ..2 Cmmm k C2z 4 j 4 k ..2 Cmme g 8 l 1 * C2x2yz 22 F222 4 a 222 Fmmm a F 4 b 4 c 4 d 8 e 2.. Fmmm g F2x 8 j 8 f .2. F2y 8 i 8 g ..2 F2z 8 h 16 k 1 * F2x2yz 23 I222 2 a 222 Immm a I 2 b 2 c 2 d 4 e 2.. Immm e I2x 4 f 4 g .2. I2y 4 h 4 i ..2 I2z 4 j 8 k 1 * I2x2yz 24 4 a 2.. Imma e 4 b .2. 4 c ..2 8 d 1 * 25 Pmm2 1 a mm2 Pmmm a P[z] 1 b 1 c 1 d 2 e .m. Pmmm i P2x[z] 2 f 2 g m.. P2y[z] 2 h 4 i 1 Pmmm u P2x2y[z] 26 2 a m.. Pmma e 2 b 4 c 1 Pmma k 27 Pcc2 2 a ..2 Pmmm a 2 b 2 c 2 d 4 e 1 Pccm q 28 Pma2 2 a ..2 Pmmm a 2 b 2 c m.. Pmma e 4 d 1 Pmma i 29 4 a 1 Pbcm d 30 Pnc2 2 a ..2 Cmmm a A[z] 2 b 4 c 1 Pmna h 2.. A2xy[z] 31 2 a m.. Pmmn a 4 b 1 Pmmn e 32 Pba2 2 a ..2 Cmmm a C[z] 2 b 4 c 1 Pbam g b.. C2xy[z] 33 4 a 1 Pnma c 34 Pnn2 2 a ..2 Immm a I[z] 2 b 4 c 1 Pnnm g n.. I2xy[z] 35 Cmm2 2 a mm2 Cmmm a C[z] 2 b 4 c ..2 Pmmm a 4 d .m. Cmmm g C2x[z] 4 e m.. C2y[z] 8 f 1 Cmmm p C2x2y[z] 36 4 a m.. Cmcm c 8 b 1 Cmcm g 37 Ccc2 4 a ..2 Cmmm a 4 b 4 c ..2 Fmmm a 8 d 1 Cccm l 38 Amm2 2 a mm2 Cmmm a A[z] 2 b 4 c .m. Cmmm k A2x[z] 4 d m.. Cmmm g A2y[z] 4 e 8 f 1 Cmmm n A2x2y[z] 39 Aem2 4 a ..2 Pmmm a 4 b 4 c .m. Cmme g 8 d 1 Cmme m 40 Ama2 4 a ..2 Cmmm a 4 b m.. Cmcm c 8 c 1 Cmcm f 41 Aea2 4 a ..2 Fmmm a F[z] 8 b 1 Cmce f .2. F2xy[z] 42 Fmm2 4 a mm2 Fmmm a F[z] 8 b ..2 Pmmm a 8 c m.. Fmmm g F2y[z] 8 d .m. F2x[z] 16 e 1 Fmmm m F2x2y[z] 43 Fdd2 8 a ..2 Fddd a D[z] 16 b 1 * d.. D2xy[z] 44 Imm2 2 a mm2 Immm a I[z] 2 b 4 c .m. Immm e I2x[z] 4 d m.. I2y[z] 8 e 1 Immm l I2x2y[z] 45 Iba2 4 a ..2 Cmmm a 4 b 8 c 1 Ibam j 46 Ima2 4 a ..2 Cmmm a 4 b m.. Imma e 8 c 1 Imma h 47 Pmmm 1 a mmm * P 1 b 1 c 1 d 1 e 1 f 1 g 1 h 2 i 2mm * P2x 2 j 2 k 2 l 2 m m2m P2y 2 n 2 o 2 p 2 q mm2 P2z 2 r 2 s 2 t 4 u m.. * P2y2z 4 v 4 w .m. P2x2z 4 x 4 y ..m P2x2y 4 z 8 α 1 * P2x2y2z 48 Pnnn 2 a 222 Immm a I 2 b 2 c 2 d 4 e Fmmm a 4 f 4 g 2.. Immm e I2x 4 h 4 i .2. I2y 4 j 4 k ..2 I2z 4 l 8 m 1 * n.. I2x2yz 49 Pccm 2 a Pmmm a 2 b 2 c 2 d 2 e 222 Pmmm a 2 f 2 g 2 h 4 i 2.. Pmmm i 4 j 4 k .2. 4 l 4 m ..2 Pmmm i 4 n 4 o 4 p 4 q ..m * 8 r 1 * 50 Pban 2 a 222 Cmmm a C 2 b 2 c 2 d 4 e Pmmm a 4 f 4 g 2.. Cmmm g C2x 4 h 4 i .2. C2y 4 j 4 k ..2 Cmmm k C2z 4 l 8 m 1 * b.. C2x2yz 51 Pmma 2 a Pmmm a 2 b 2 c 2 d 2 e mm2 * 2 f 4 g .2. Pmmm i 4 h 4 i .m. * 4 j 4 k m.. * 8 l 1 * 52 Pnna 4 a Cmmm a 4 b 4 c ..2 Imma e 4 d 2.. Cmcm c 8 e 1 * 53 Pmna 2 a Cmmm a B 2 b 2 c 2 d 4 e 2.. Cmmm g B2x 4 f 4 g .2. Pmma e 4 h m.. * .2. B2yz 8 i 1 * .2. B2yz2x 54 Pcca 4 a Pmmm a 4 b 4 c .2. Cmme g 4 d ..2 Pmma e 4 e 8 f 1 * 55 Pbam 2 a Cmmm a C 2 b 2 c 2 d 4 e ..2 Cmmm k C2z 4 f 4 g ..m * b.. C2xy 4 h 8 i 1 * b.. C2xy2z 56 Pccn 4 a Fmmm a F 4 b 4 c ..2 Pmmn a 4 d 8 e 1 * c.2 F2xyz 57 Pbcm 4 a Pmmm a 4 b 4 c 2.. Pmma e 4 d ..m * 8 e 1 * 58 Pnnm 2 a Immm a I 2 b 2 c 2 d 4 e ..2 Immm e I2z 4 f 4 g ..m * n.. I2xy 8 h 1 * n.. I2xy2z 59 Pmmn 2 a mm2 * 2 b 4 c Pmmm a 4 d 4 e m.. * 4 f .m. 8 g 1 * 60 Pbcn 4 a Cmmm a 4 b 4 c .2. Cmcm c 8 d 1 * 61 Pbca 4 a Fmmm a F 4 b 8 c 1 * bc. F2xyz 62 Pnma 4 a Cmmm a 4 b 4 c .m. * 8 d 1 * 63 Cmcm 4 a Cmmm a 4 b 4 c m2m * 8 d Pmmm a 8 e 2.. Cmmm g 8 f m.. * 8 g ..m * 16 h 1 * 64 Cmce 4 a Fmmm a F 4 b 8 c Pmmm a 8 d 2.. Fmmm g F2x 8 e .2. Pmma e 8 f m.. * .2. F2yz 16 g 1 * .2. F2yz2x 65 Cmmm 2 a mmm * C 2 b 2 c 2 d 4 e Pmmm a 4 f 4 g 2mm * C2x 4 h 4 i m2m C2y 4 j 4 k mm2 * C2z 4 l 8 m ..2 Pmmm i 8 n m.. * C2y2z 8 o .m. C2x2z 8 p ..m * C2x2y 8 q 16 r 1 * C2x2y2z 66 Cccm 4 a 222 Cmmm a 4 b 4 c Cmmm a 4 d 4 e Fmmm a 4 f 8 g 2.. Cmmm g 8 h .2. 8 i ..2 Cmmm k 8 j 8 k ..2 Fmmm g 8 l ..m * 16 m 1 * 67 Cmme 4 a 222 Pmmm a 4 b 4 c Pmmm a 4 d 4 e 4 f 4 g mm2 * 8 h 2.. Pmmm i 8 i 8 j .2. 8 k 8 l ..2 Pmmm i 8 m m.. * 8 n .m. 16 o 1 * 68 Ccce 4 a 222 Fmmm a F 4 b 8 c Pmmm a 8 d 8 e 2.. Fmmm g F2x 8 f .2. F2y 8 g ..2 Fmmm g F2z 8 h ..2 Cmme g 16 i 1 * c.. F2x2yz 69 Fmmm 4 a mmm * F 4 b 8 c Pmmm a 8 d 8 e 8 f 222 Pmmm a 8 g 2mm * F2x 8 h m2m F2y 8 i mm2 F2z 16 j ..2 Pmmm i 16 k .2. 16 l 2.. 16 m m.. * F2y2z 16 n .m. F2x2z 16 o ..m F2x2y 32 p 1 * F2x2y2z 70 Fddd 8 a 222 * D 8 b 16 c * T 16 d 16 e 2.. * D2x 16 f .2. D2y 16 g ..2 D2z 32 h 1 * d.. D2x2yz 71 Immm 2 a mmm * I 2 b 2 c 2 d 4 e 2mm * I2x 4 f 4 g m2m I2y 4 h 4 i mm2 I2z 4 j 8 k Pmmm a 8 l m.. * I2y2z 8 m .m. I2x2z 8 n ..m I2x2y 16 o 1 * I2x2y2z 72 Ibam 4 a 222 Cmmm a 4 b 4 c Cmmm a 4 d 8 e Pmmm a 8 f 2.. Cmmm g 8 g .2. 8 h ..2 Cmmm k 8 i 8 j ..m * 16 k 1 * 73 Ibca 8 a Pmmm a 8 b 8 c 2.. Cmme g 8 d .2. 8 e ..2 16 f 1 * 74 Imma 4 a Cmmm a 4 b 4 c 4 d 4 e mm2 * 8 f 2.. Cmmm g 8 g .2. 8 h m.. * 8 i .m. 16 j 1 * 75 P4 1 a 4.. P[z] 1 b 2 c 2.. 4 d 1 P4xy[z] 76 4 a 1 * 77 2 a 2.. 2 b 2 c 2.. 4 d 1 78 4 a * 79 I4 2 a 4.. I[z] 4 b 2.. 8 c 1 I4xy[z] 80 4 a 2.. 8 b 1 * 81 1 a P 1 b 1 c 1 d 2 e 2.. P2z 2 f 2 g 2.. 4 h 1 * P4xyz 82 2 a I 2 b 2 c 2 d 4 e 2.. I2z 4 f 8 g 1 * I4xyz 83 1 a P 1 b 1 c 1 d 2 e 2 f 2 g 4.. P2z 2 h 4 i 2.. 4 j m.. * P4xy 4 k 8 l 1 * P4xy2z 84 2 a 2 b 2 c 2 d 2 e 2 f 4 g 2.. 4 h 4 i 2.. 4 j m.. * 8 k 1 * 85 2 a C 2 b 2 c 4.. 4 d 4 e 4 f 2.. C2z 8 g 1 * 86 2 a I 2 b 4 c 4 d 4 e 2.. 4 f 2.. I2z 8 g 1 * n.. I4xyz 87 2 a I 2 b 4 c 4 d 4 e 4.. I2z 8 f 8 g 2.. 8 h m.. * I4xy 16 i 1 * I4xy2z 88 4 a 4 b 8 c 8 d 8 e 2.. 16 f 1 * 89 P422 1 a 422 P 1 b 1 c 1 d 2 e 222. 2 f 2 g 4.. P2z 2 h 4 i 2.. 4 j ..2 P4xx 4 k 4 l .2. P4x 4 m 4 n 4 o 8 p 1 * P4x2yz 90 2 a 2.22 C 2 b 2 c 4.. 4 d 2.. C2z 4 e ..2 .b. C2xx 4 f 8 g 1 * 91 4 a .2. * 4 b 4 c ..2 * 8 d 1 * 92 4 a ..2 * 8 b 1 * 93 2 a 222. 2 b 2 c 222. 2 d 2 e 2.22 2 f 4 g 2.. 4 h 4 i 2.. 4 j .2. 4 k 4 l 4 m 4 n ..2 4 o 8 p 1 * 94 2 a 2.22 I 2 b 4 c 2.. I2z 4 d 2.. 4 e ..2 .n. I2xx 4 f 8 g 1 * 95 4 a .2. * 4 b 4 c ..2 * 8 d 1 * 96 4 a ..2 * 8 b 1 * 97 I422 2 a 422 I 2 b 4 c 222. 4 d 2.22 4 e 4.. I2z 8 f 2.. 8 g ..2 I4xx 8 h .2. I4x 8 i 8 j ..2 16 k 1 * I4x2yz 98 4 a 2.22 4 b 8 c 2.. 8 d ..2 * 8 e 8 f .2. * 16 g 1 * 99 P4mm 1 a 4mm P[z] 1 b 2 c 2mm. 4 d ..m P4xx[z] 4 e .m. P4x[z] 4 f 8 g 1 P4x2y[z] 100 P4bm 2 a 4.. C[z] 2 b 2.mm 4 c ..m 8 d 1 ..m C4xy[z] 101 2 a 2.mm 2 b 4 c 2.. 4 d ..m 8 e 1 102 2 a 2.mm I[z] 4 b 2.. 4 c ..m .n. I2xx[z] 8 d 1 .n. I2xx2y[z] 103 P4cc 2 a 4.. 2 b 4 c 2.. 8 d 1 104 P4nc 2 a 4.. I[z] 4 b 2.. 8 c 1 ..2 I4xy[z] 105 2 a 2mm. 2 b 2 c 2mm. 4 d .m. 4 e 8 f 1 106 4 a 2.. 4 b 2.. 8 c 1 107 I4mm 2 a 4mm I[z] 4 b 2mm. 8 c ..m I4xx[z] 8 d .m. I4x[z] 16 e 1 I4x2y[z] 108 I4cm 4 a 4.. 4 b 2.mm 8 c ..m 16 d 1 109 4 a 2mm. 8 b .m. * 16 c 1 * 110 8 a 2.. 16 b 1 * 111 1 a P 1 b 1 c 1 d 2 e 222. 2 f 2 g 2.mm P2z 2 h 4 i .2. P4x 4 j 4 k 4 l 4 m 2.. 4 n ..m * P4xxz 8 o 1 * P4xxz2y 112 2 a 222. 2 c 2 b 222. 2 d 2 e 2 f 4 g .2. 4 h 4 i 4 j 4 k 2.. 4 l 4 m 2.. 8 n 1 * 113 2 a C 2 b 2 c 2.mm 4 d 2.. C2z 4 e ..m * 8 f 1 * ..m C4xyz 114 2 a I 2 b 4 c 2.. I2z 4 d 2.. 8 e 1 * ..c I4xyz 115 1 a P 1 b 1 c 1 d 2 e 2mm. P2z 2 f 2 g 2mm. 4 h ..2 P4xx 4 i 4 j .m. * P4xz 4 k 8 l 1 * P4xz2y 116 2 a 2.22 2 b 2 c 2 d 4 e ..2 4 f 4 g 2.. 4 h 4 i 2.. 8 j 1 * 117 2 a C 2 b 2 c 2.22 2 d 4 e 2.. C2z 4 f 2.. 4 g ..2 4 h 8 i 1 * ..2 C4xyz 118 2 a I 2 b 2 c 2.22 2 d 4 e 2.. I2z 4 f ..2 4 g 4 h 2.. 8 i 1 * ..2 I4xyz 119 2 a I 2 b 2 c 2 d 4 e 2mm. I2z 4 f 8 g ..2 I4xx 8 h 8 i .m. * I4xz 16 j 1 * I4xz2y 120 4 a 2.22 4 d 4 b 4 c 8 e ..2 8 h 8 f 2.. 8 g 16 i 1 * 121 2 a I 2 b 4 c 222. 4 d 4 e 2.mm I2z 8 f .2. I4x 8 g 8 h 2.. 8 i ..m * I4xxz 16 j 1 * I4xxz2y 122 4 a 4 b 8 c 2.. 8 d .2. * 16 e 1 * 123 1 a * P 1 b 1 c 1 d 2 e mmm. 2 f 2 g 4mm * P2z 2 h 4 i 2mm. 4 j m.2m * P4xx 4 k 4 l m2m. * P4x 4 m 4 n 4 o 8 p m.. * P4x2y 8 q 8 r ..m * P4xx2z 8 s .m. * P4x2z 8 t 16 u 1 * P4x2y2z 124 2 a 422 2 c 2 b 2 d 4 e 4 f 222. 4 g 4.. 4 h 8 i 2.. 8 j ..2 8 k .2. 8 l 8 m m.. * 16 n 1 * 125 2 a 422 C 2 b 2 c 2 d 4 e 4 f 4 g 4.. C2z 4 h 2.mm 8 i ..2 C4xx 8 j 8 k .2. C4x 8 l 8 m ..m * 16 n 1 * ..m C4x2yz 126 2 a 422 I 2 b 4 c 222. 4 d 4 e 4.. I2z 8 f 8 g 2.. 8 h ..2 I4xx 8 i .2. I4x 8 j 16 k