International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 745-769

Table 3.2.3.2 

H. Klapper,a Th. Hahna and M. I. Aroyoc

Table 3.2.3.2| top | pdf |
The 32 three-dimensional crystallographic point groups

The point groups are listed in blocks according to crystal system and are specified by their short and (if different) full Hermann–Mauguin symbols and their Schoenflies symbols. For each point group, the stereographic projections show (on the left) the general position and (on the right) the symmetry elements. The list of Wyckoff positions includes: Columns 1 to 4: multiplicity, Wyckoff letter, oriented site-symmetry symbol, coordinate triplets; Under the stereographic projections: face forms (in roman type) and point forms (in italics); if there is more than one entry, subsequent entries refer to limiting (noncharacteristic) forms; Last column: Miller indices of equivalent faces [for trigonal and hexagonal groups, Bravais–Miller indices (hkil) are used if referred to hexagonal axes].

TRICLINIC SYSTEM
1 [C_{1}] [Scheme scheme12]  
1 a 1 [x,y,z] Pedion or monohedron (hkl)
        Single point  
        Symmetry of special projections  
        Along any direction  
        1  
[\bar{1}] [C_{i}] [Scheme scheme13]  
2 a 1 [x,y,z\quad \bar x,\bar y,\bar z] Pinacoid or parallelohedron [(hkl) \quad(\bar{h}\bar{k}\bar{l})]
        Line segment through origin  
1 o [\bar 1] [0,0,0] Point in origin  
        Symmetry of special projections  
        Along any direction  
        2  
MONOCLINIC SYSTEM
2 [C_{2}] [Scheme scheme14]  
UNIQUE AXIS b  
2 b 1 [x,y,z\quad \bar x,y,\bar z] Sphenoid or dihedron [(hkl) \quad(\bar{h}k\bar{l})]
        Line segment  
        Pinacoid or parallelohedron [(h0l) \quad(\bar{h}0\bar{l})]
        Line segment through origin  
1 a 2 [0,y,0] Pedion or monohedron [(010) \hbox{ or } (0\bar{1}0)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&2&&m\cr}]  
2 [C_{2}] [Scheme scheme15]  
UNIQUE AXIS c  
2 b 1 [x,y,z \quad \bar x,\bar y,z] Sphenoid or dihedron [(hkl) \quad(\bar{h}\bar{k}l)]
        Line segment  
        Pinacoid or parallelohedron [(hk0) \quad(\bar{h}\bar{k}0)]
        Line segment through origin  
1 a 2 [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&m&&2}]  
m [C_{s}] [Scheme scheme16]  
UNIQUE AXIS b  
2 b 1 [x,y,z\quad x,\bar y,z] Dome or dihedron [(hkl) \quad(h\bar{k}l)]
        Line segment  
        Pinacoid or parallelohedron [(010) \quad(0\bar{1}0)]
        Line segment through origin  
1 a m [x,0,z ] Pedion or monohedron [(h0l)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&1&&m\cr}]  
m [C_{s}] [Scheme scheme17]  
UNIQUE AXIS c  
2 b 1 [x,y,z\quad x,y,\bar z] Dome or dihedron [(hkl) \quad(hk\bar{l})]
        Line segment  
        Pinacoid or parallelohedron [(001) \quad(00\bar{1})]
        Line segment through origin  
1 a m [x,y,0] Pedion or monohedron [(hk0)]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along\ [100]}&&\hbox{ Along\ [010]}&&\hbox{Along\ [001]}\cr m&&m&&1\cr}]  
2/m [C_{2h}] [Scheme scheme18]  
UNIQUE AXIS b  
4 c 1 [x,y,z\quad \bar x,y,\bar z\quad \bar x,\bar y,\bar z\quad x,\bar y,z] Rhombic prism [(hkl) \quad(\bar{h}k\bar{l}) \quad(\bar{h}\bar{k}\bar{l}) \quad(h\bar{k}l)]
        Rectangle through origin  
2 b m [x,0,z\quad \bar x,0,\bar z] Pinacoid or parallelohedron [(h0l) \quad(\bar{h}0\bar{l})]
        Line segment through origin  
2 a 2 [0,y,0\quad 0,\bar y,0] Pinacoid or parallelohedron [(010) \quad(0\bar{1}0)]
        Line segment through origin  
1 o 2/m [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along } [100]&&\hbox{Along } [010]&&\hbox{Along }[001]\cr 2mm&&2&&2mm\cr }]  
2/m [C_{2h}] [Scheme scheme19]  
UNIQUE AXIS c  
4 c 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,\bar y,\bar z\quad x,y,\bar z] Rhombic prism [(hkl) \quad(\bar{h}\bar{k}l) \quad(\bar{h}\bar{k}\bar{l}) \quad(hk\bar{l})]
        Rectangle through origin  
2 b m [x,y,0\quad \bar x,\bar y,0] Pinacoid or parallelohedron [(hk0) \quad(\bar{h}\bar{k}0)]
        Line segment through origin  
2 a 2 [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [(001) \quad(00\bar{1})]
        Line segment through origin  
1 o 2/m [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along } [100]&&\hbox{Along } [010]&&\hbox{Along }[001]\cr 2mm&&2mm&&2}]  
ORTHORHOMBIC SYSTEM
222 [D_{2}] [Scheme scheme20]  
4 d 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z] Rhombic disphenoid or rhombic tetrahedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]
        Rhombic tetrahedron  
        Rhombic prism [\matrix{(hk0) {\hbox to 7.5pt{}}(\bar{h}\bar{k}0) {\hbox to 7.5pt{}}(\bar{h}k0) {\hbox to 9pt{}}(h\bar{k}0)\cr}]
        Rectangle through origin  
        Rhombic prism [\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]
        Rectangle through origin  
        Rhombic prism [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]
        Rectangle through origin  
2 c ..2 [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(001) {\hbox to 7.5pt{}}(00\bar{1})\cr}]
        Line segment through origin  
2 b .2. [0,y,0\quad 0,\bar y,0] Pinacoid or parallelohedron [\matrix{(010) {\hbox to 7.5pt{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 a 2.. [x,0,0\quad \bar x,0,0] Pinacoid or parallelohedron [\matrix{(100) {\hbox to 7.5pt{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 o 222 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr 2mm&&2mm&&2mm\cr}]  
mm2 [C_{2v}] [Scheme scheme21]  
4 d 1 [x,y,z\quad \bar x,\bar y,z\quad x,\bar y,z\quad \bar x,y,z] Rhombic pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(h\bar{k}l) &(\bar{h}kl)\cr}]
        Rectangle  
        Rhombic prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0)\cr}]
        Rectangle through origin  
2 c m.. [0,y,z\quad 0,\bar y,z] Dome or dihedron [\matrix{(0kl) &(0\bar{k}l)\cr}]
        Line segment  
        Pinacoid or parallelohedron [\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 b .m. [x,0,z\quad \bar x,0,z] Dome or dihedron [\matrix{(h0l) &(\bar{h}0l)\cr}]
        Line segment  
        Pinacoid or parallelohedron [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 a mm2 [0,0,z ] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr m&&m&&2mm\cr}]  
[\openup 6pt\matrix{m\ m\ m\cr{\displaystyle{2 \over m}\ {2 \over m}\ {2 \over m}}\cr}] [D_{2h}] [Scheme scheme22]  
8 g 1 [x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z] Rhombic dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr}]
      [\bar x,\bar y,\bar z\quad x,y,\bar z\quad x,\bar y,z\quad \bar x,y,z] Rectangular prism [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl)\cr}]
4 f ..m [x,y,0\quad \bar x,\bar y,0\quad \bar x,y,0\quad x,\bar y,0] Rhombic prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0)\cr}]
        Rectangle through origin  
4 e .m. [x,0,z\quad \bar x,0,z\quad \bar x,0,\bar z\quad x,0,\bar z] Rhombic prism [\matrix{(h0l) &(\bar{h}0l) &(\bar{h}0\bar{l}) &(h0\bar{l})\cr}]
        Rectangle through origin  
4 d m.. [0,y,z\quad 0,\bar y,z\quad 0,y,\bar z\quad 0,\bar y,\bar z] Rhombic prism [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr}]
        Rectangle through origin  
2 c mm2 [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
2 b m2m [0,y,0\quad 0,\bar y,0 ] Pinacoid or parallelohedron [\matrix{(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Line segment through origin  
2 a 2mm [x,0,0\quad \bar x,0,0] Pinacoid or parallelohedron [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00)\cr}]
        Line segment through origin  
1 o mmm [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[100]&&\hbox{Along }[010]&&\hbox{Along }[001]\cr 2mm&&2mm&&2mm\cr}]  
TETRAGONAL SYSTEM
4 [C_{4}] [Scheme scheme23]  
4 b 1 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
        Square through origin  
1 a 4.. [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&m&&m\cr}]  
[\bar{4}] [S_{4}] [Scheme scheme24]  
4 b 1 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
        Tetragonal tetrahedron  
        Tetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(k\bar{h}0) {\hbox to .65pc{}}(\bar{k}h0)\cr}]
        Square through origin  
2 a 2.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 4..] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&m&&m\cr}]  
4/m [C_{4h}] [Scheme scheme25]  
8 c 1 [x,y,z \quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,\bar y,\bar z\quad x,y,\bar z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal prism [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
4 b m.. [x,y,0\quad \bar x,\bar y,0\quad \bar y,x,0\quad y,\bar x,0] Tetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
        Square through origin  
2 a 4.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 4/m.. [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4&&2mm&&2mm\cr}]  
422 [D_{4}] [Scheme scheme26]  
8 d 1 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Tetragonal trapezohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,y,\bar z\quad x,\bar y,\bar z\quad y,x,\bar z\quad \bar y,\bar x,\bar z] Twisted tetragonal antiprism [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]
        Ditetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
        Truncated square through origin [\matrix{(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .65pc{}}(kh0) {\hbox to .65pc{}}(\bar{k}\bar{h}0)\cr}]
        Tetragonal dipyramid [\matrix{(h0l) {\hbox to .8pc{}}(\bar{h}0l) {\hbox to .8pc{}}(0hl) {\hbox to .8pc{}}(0\bar{h}l)\cr}]
        Tetragonal prism [\matrix{(\bar{h}0\bar{l}) {\hbox to .8pc{}}(h0\bar{l}) {\hbox to .8pc{}}(0h\bar{l}) {\hbox to .8pc{}}(0\bar{h}\bar{l})\cr}]
        Tetragonal dipyramid [\matrix{(hhl) {\hbox to .8pc{}}(\bar{h}\bar{h}l) {\hbox to .8pc{}}(\bar{h}hl) {\hbox to .8pc{}}(h\bar{h}l)\cr}]
        Tetragonal prism [\matrix{(\bar{h}h\bar{l}) {\hbox to .8pc{}}(h\bar{h}\bar{l}) {\hbox to .8pc{}}(hh\bar{l}) {\hbox to .8pc{}}(\bar{h}\bar{h}\bar{l})\cr}]
4 c .2. [x,0,0\quad \bar x,0,0\quad 0,x,0\quad 0,\bar x,0] Tetragonal prism [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .6pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Square through origin  
4 b ..2 [x,x,0\quad \bar x,\bar x,0\quad \bar x,x,0\quad x,\bar x,0] Tetragonal prism [\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .6pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]
        Square through origin  
2 a 4.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 422 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&2mm&&2mm\cr}]  
4mm [C_{4v}] [Scheme scheme27]  
8 d 1 [x,y,z\quad \bar x,\bar y,z\quad \bar y,x,z\quad y,\bar x,z] Ditetragonal pyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [x,\bar y,z\quad \bar x,y,z\quad \bar y,\bar x,z\quad y,x,z] Truncated square [\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]
        Ditetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
        Truncated square through origin [\matrix{(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(\bar{k}\bar{h}0) {\hbox to .65pc{}}(kh0)\cr}]
4 c .m. [x,0,z\quad \bar x,0,z\quad 0,x,z\quad 0,\bar x,z] Tetragonal pyramid [\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .65pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Square through origin  
4 b ..m [x,x,z\quad \bar x,\bar x,z\quad \bar x,x,z\quad x,\bar x,z] Tetragonal pyramid [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]
        Square  
        Tetragonal prism [\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .65pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]
        Square through origin  
1 a 4mm [0,0,z] Pedion or monohedron [(001) \hbox{ or } (00\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&m&&m\cr}]  
[\bar{4}2m] [D_{2d}] [Scheme scheme28]  
8 d 1 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal scalenohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [\bar x,y,\bar z\quad x,\bar y,\bar z\quad \bar y,\bar x,z\quad y,x,z] Tetragonal tetrahedron cut off by pinacoid [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(\bar{k}\bar{h}l) &(khl)\cr}]
        Ditetragonal prism [\matrix{(hk0) {\hbox to 0.65pc{}}(\bar{h}\bar{k}0) {\hbox to 0.65pc{}}(k\bar{h}0) {\hbox to 0.7pc{}}(\bar{k}h0)\cr}]
        Truncated square through origin [\matrix{(\bar{h}k0) {\hbox to 0.65pc{}}(h\bar{k}0) {\hbox to 0.65pc{}}(\bar{k}\bar{h}0) {\hbox to 0.65pc{}}({\it kh}0)\cr}]
        Tetragonal dipyramid [\matrix{(h0l) &(\bar{h}0l) {\hbox to 0.75pc{}}(0\bar{h}\bar{l}) {\hbox to 0.85pc{}}(0h\bar{l})\cr}]
        Tetragonal prism [\matrix{(\bar{h}0\bar{l}) {\hbox to 0.8pc{}}(h0\bar{l}) {\hbox to 0.8pc{}}(0\bar{h}l) {\hbox to 0.8pc{}}(0hl)\cr}]
4 c ..m [x,x,z\quad \bar x,\bar x,z\quad x,\bar x,\bar z\quad \bar x,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\matrix{(hhl) {\hbox to 0.8pc{}}(\bar{h}\bar{h}l) {\hbox to 0.8pc{}}(h\bar{h}\bar{l}) {\hbox to 0.8pc{}}(\bar{h}h\bar{l})\cr}]
        Tetragonal tetrahedron  
        Tetragonal prism [\matrix{(110) {\hbox to 0.65pc{}}(\bar{1}\bar{1}0) {\hbox to 0.65pc{}}(1\bar{1}0) {\hbox to 0.65pc{}}(\bar{1}10)\cr}]
        Square through origin  
4 b .2. [x,0,0\quad \bar x,0,0\quad 0,\bar x,0\quad 0,x,0] Tetragonal prism [\matrix{(100) {\hbox to 0.65pc{}}(\bar{1}00) {\hbox to 0.65pc{}}(0\bar{1}0) {\hbox to 0.65pc{}}(010)\cr}]
        Square through origin  
2 a 2.mm [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to 0.65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 42m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&2mm&&m\cr}]  
[\bar{4}m2] [D_{2d}] [Scheme scheme29]  
8 d 1 [x,y,z\quad \bar x,\bar y,z\quad y,\bar x,\bar z\quad \bar y,x,\bar z] Tetragonal scalenohedron [\matrix{(hkl) &(\bar{h}\bar{k}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [x,\bar y,z\quad \bar x,y,z\quad y,x,\bar z\quad \bar y,\bar x,\bar z] Tetragonal tetrahedron cut off by pinacoid [\matrix{(h\bar{k}l) {\hbox to .8pc{}}(\bar{h}kl) {\hbox to .85pc{}}(kh\bar{l}) {\hbox to .9pc{}}(\bar{k}\bar{h}\bar{l})\cr}]
        Ditetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .65pc{}}(k\bar{h}0) {\hbox to .7pc{}}(\bar{k}h0)\cr}]
        Truncated square through origin [\matrix{(h\bar{k}0) {\hbox to .65pc{}}(\bar{h}k0) {\hbox to .65pc{}}(kh0) {\hbox to .7pc{}}(\bar{k}\bar{h}0)\cr}]
        Tetragonal dipyramid [\matrix{(hhl) {\hbox to .75pc{}}(\bar{h}\bar{h}l) {\hbox to .85pc{}}(h\bar{h}\bar{l}) {\hbox to .85pc{}}(\bar{h}h\bar{l})\cr}]
        Tetragonal prism [\matrix{(h\bar{h}l) {\hbox to .8pc{}}(\bar{h}hl) {\hbox to .8pc{}}(hh\bar{l}) {\hbox to .85pc{}}(\bar{h}\bar{h}\bar{l})\cr}]
4 c .m. [x,0,z\quad \bar x,0,z\quad 0,\bar x,\bar z\quad 0,x,\bar z] Tetragonal disphenoid or tetragonal tetrahedron [\matrix{(h0l) {\hbox to .8pc{}}(\bar{h}0l) {\hbox to .8pc{}}(0\bar{h}\bar{l}) {\hbox to .85pc{}}(0h\bar{l})\cr}]
        Tetragonal tetrahedron  
        Tetragonal prism [\matrix{(100) &{\hbox to -2pt{}}(\bar{1}00) &{\hbox to -3pt{}}(0\bar{1}0) &{\hbox to -1.5pt{}}(010)\cr}]
        Square through origin  
4 b ..2 [x,x,0\quad \bar x,\bar x,0\quad x,\bar x,0\quad \bar x,x,0] Tetragonal prism [\matrix{(110) &{\hbox to -2pt{}}(\bar{1}\bar{1}0) &{\hbox to -3pt{}}(1\bar{1}0) &{\hbox to -1.5pt{}}(\bar{1}10)\cr}]
        Square through origin  
2 a 2mm. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) &{\hbox to -2pt{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 4 m2] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[110]\cr 4mm&&m&&2mm\cr}]  
[\openup6pt\matrix{4/mmm\hfill\cr{\displaystyle{4 \over m}{2 \over m}{2 \over m}}\hfill}] [D_{4h}] [Scheme scheme30]  
16 g 1 [x,y,z \quad \bar x,\bar y,z \quad \bar y,x,z \quad y,\bar x,z] Ditetragonal dipyramid [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{k}hl) &(k\bar{h}l)\cr}]
      [\bar x,y,\bar z \quad x,\bar y,\bar z \quad y,x,\bar z \quad \bar y,\bar x,\bar z] Edge-truncated tetragonal prism [\matrix{(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) &(kh\bar{l}) &(\bar{k}\bar{h}\bar{l})\cr}]
      [\bar x,\bar y,\bar z \quad x,y,\bar z \quad y,\bar x,\bar z \quad \bar y,x,\bar z]   [\matrix{(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr}]
      [x,\bar y,z \quad \bar x,y,z \quad \bar y,\bar x,z \quad y,x,z]   [\matrix{(h\bar{k}l) &(\bar{h}kl) &(\bar{k}\bar{h}l) &(khl)\cr}]
8 f .m. [x,0,z\quad \bar x,0,z\quad 0,x,z\quad 0,\bar x,z] Tetragonal dipyramid [\matrix{(h0l) &(\bar{h}0l) &(0hl) &(0\bar{h}l)\cr}]
      [\bar x,0,\bar z\quad x,0,\bar z\quad 0,x,\bar z\quad 0,\bar x,\bar z] Tetragonal prism [\matrix{(\bar{h}0\bar{l}) &(h0\bar{l}) &(0h\bar{l}) &(0\bar{h}\bar{l})\cr}]
8 e ..m [x,x,z\quad \bar x,\bar x,z\quad \bar x,x,z\quad x,\bar x,z] Tetragonal dipyramid [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}hl) &(h\bar{h}l)\cr}]
      [\bar x,x,\bar z\quad x,\bar x,\bar z\quad x,x,\bar z\quad \bar x,\bar x,\bar z] Tetragonal prism [\matrix{(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) &(hh\bar{l}) &(\bar{h}\bar{h}\bar{l})\cr}]
8 d m.. [x,y,0\quad \bar x,\bar y,0\quad \bar y,x,0\quad y,\bar x,0] Ditetragonal prism [\matrix{(hk0) {\hbox to .65pc{}}(\bar{h}\bar{k}0) {\hbox to .7pc{}}(\bar{k}h0) {\hbox to .65pc{}}(k\bar{h}0)\cr}]
      [\bar x,y,0\quad x,\bar y,0\quad y,x,0\quad \bar y,\bar x,0] Truncated square through origin [\matrix{(\bar{h}k0) {\hbox to .65pc{}}(h\bar{k}0) {\hbox to .7pc{}}(kh0) {\hbox to .65pc{}}(\bar{k}\bar{h}0)\cr}]
4 c m2m. [x,0,0\quad \bar x,0,0\quad 0,x,0\quad 0,\bar x,0] Tetragonal prism [\matrix{(100) {\hbox to .65pc{}}(\bar{1}00) {\hbox to .7pc{}}(010) {\hbox to .65pc{}}(0\bar{1}0)\cr}]
        Square through origin  
4 b m.m2 [x,x,0\quad \bar x,\bar x,0\quad \bar x,x,0\quad x,\bar x,0] Tetragonal prism [\matrix{(110) {\hbox to .65pc{}}(\bar{1}\bar{1}0) {\hbox to .7pc{}}(\bar{1}10) {\hbox to .65pc{}}(1\bar{1}0)\cr}]
        Square through origin  
2 a 4mm [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(001) {\hbox to .65pc{}}(00\bar{1})\cr}]
        Line segment through origin  
1 o 4/mmm [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox {Along }[100]&&\hbox{\ Along }[110]\cr 4mm&&2mm&&2mm\cr}]  
TRIGONAL SYSTEM
3 [C_{3}] [Scheme scheme31]  
HEXAGONAL AXES  
3 b 1 [x,y,z \quad \bar y,x-y,z \quad \bar x+y,\bar x,z] Trigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
        Trigon  
        Trigonal prism [\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]
        Trigon through origin  
1 a 3.. [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
      Symmetry of special projections
      [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3&&1&&1\cr}]
3 [C_{3}] [Scheme scheme32]  
RHOMBOHEDRAL AXES  
3 b 1 [x,y,z\quad z,x,y\quad y,z,x] Trigonal pyramid [\matrix{(hkl) &(lhk) &(klh)\cr}]
        Trigon  
        Trigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Trigon through origin  
1 a 3. [x,x,x] Pedion or monohedron [(111) \hbox{ or } (\bar{1}\bar{1}\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3&&1&&1\cr}]  
[\bar{3}] [C_{3i}] [Scheme scheme33]  
HEXAGONAL AXES  
6 b 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Rhombohedron [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z] Trigonal antiprism [\matrix{(\bar{h}\bar{k}\bar{i}\,\!\bar{l}) {\hbox to .8pc{}}(\bar{i}\bar{h}\bar{k}\bar{l}) {\hbox to .8pc{}}(\bar{k}\bar{i}\bar{h}\bar{l})\cr}]
        Hexagonal prism [\matrix{(hki0) {\hbox to .65pc{}}(ihk0) {\hbox to .65pc{}}(kih0)\cr}]
        Hexagon through origin [\matrix{(\bar{h}\bar{k}\bar{i}0) {\hbox to .6pc{}}(\bar{i}\bar{h}\bar{k}0) {\hbox to .65pc{}}(\bar{k}\bar{i}\bar{h}0)\cr}]
2 a 3.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to .4pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3..] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&2&&2\cr}]  
[\bar{3}] [C_{3i}] [Scheme scheme34]  
RHOMBOHEDRAL AXES  
6 b 1 [x,y,z\quad z,x,y\quad y,z,x] Rhombohedron [\matrix{(hkl) &(lhk) &(klh)\cr}]
      [\bar x,\bar y,\bar z\quad \bar z,\bar x,\bar y\quad \bar y,\bar z,\bar x] Trigonal antiprism [\matrix{(\bar{h}\bar{k}\bar{l}) &(\bar{l}\bar{h}\bar{k}) &(\bar{k}\bar{l}\bar{h})\cr}]
        Hexagonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\! +\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]
        Hexagon through origin [\matrix{(\bar{h}\bar{k}(h\!+\!k)) &((h\! +\! k)\bar{h}\bar{k}) &(\bar{k}(h\! +\! k)\bar{h})\cr}]
2 a 3. [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3.] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 6&&2&&2\cr}]  
312 [D_{3}] [Scheme scheme36]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Trigonal trapezohedron [\matrix{(hkil) {\hbox to 1.5pc{}}(ihkl) {\hbox to 1.45pc{}}(kihl)\cr}]
      [\bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z] Twisted trigonal antiprism [\matrix{(\bar{k}\bar{h}\bar{i}\,\!\bar{l}) {\hbox to 1.45pc{}}(\bar{h}\bar{i}\bar{k}\bar{l}) {\hbox to 1.45pc{}}(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
        Ditrigonal prism [\matrix{(hki0) {\hbox to 1.35pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]
        Truncated trigon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) {\hbox to 1.3pc{}}(\bar{h}\bar{i}\bar{k}0) {\hbox to 1.25pc{}}(\bar{i}\bar{k}\bar{h}0)\cr}]
        Trigonal dipyramid [\matrix{(h0\bar{h}l) {\hbox to 1.3pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]
        Trigonal prism [\matrix{(0\bar{h}h\bar{l}) {\hbox to 1.3pc{}}(\bar{h}h0\bar{l}) {\hbox to 1.25pc{}}(h0\bar{h}\bar{l})\cr}]
        Rhombohedron [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Trigonal antiprism [\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) {\hbox to 1.1pc{}}(\bar{2}110) {\hbox to 1.1pc{}}(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) {\hbox to 1.1pc{}}(\bar{1}2\bar{1}0) {\hbox to 1.1pc{}}(2\bar{1}\bar{1}0)\cr}]
3 b ..2 [x,\bar x,0\quad x,2x,0\quad 2\bar x,\bar x,0] Trigonal prism [(10\bar{1}0)\quad(\bar{1}100)\quad(0\bar{1}10)]
        Trigon through origin or [(\bar{1}010)\quad(1\bar{1}00)\quad(01\bar{1}0)]
2 a 3.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to 1.1pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o 3.2 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&1&&2\cr}]  
321 [D_{3}] [Scheme scheme35]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Trigonal trapezohedron [\matrix{(hkil) {\hbox to 1.45pc{}}(ihkl) {\hbox to 1.5pc{}}(kihl)\cr}]
      [y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z] Twisted trigonal antiprism [\matrix{(khi\bar{l}) {\hbox to 1.45pc{}}(hik\bar{l}) {\hbox to 1.45pc{}}(ikh\bar{l})\cr}]
        Ditrigonal prism [\matrix{(hki0) {\hbox to 1.3pc{}}(ihk0) {\hbox to 1.25pc{}}(kih0)\cr}]
        Truncated trigon through origin [\matrix{(khi0) {\hbox to 1.3pc{}}(hik0) {\hbox to 1.3pc{}}(ikh0)\cr}]
        Trigonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Trigonal prism [\matrix{(hh\overline{2h}{\hbox to 1pt{}}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l})\cr}]
        Rhombohedron [\matrix{(h0\bar{h}l) {\hbox to 1.25pc{}}(\bar{h}h0l) {\hbox to 1.25pc{}}(0\bar{h}hl)\cr}]
        Trigonal antiprism [\matrix{(0h\bar{h}\bar{l}) {\hbox to 1.25pc{}}(h\bar{h}0\bar{l}) {\hbox to 1.25pc{}}(\bar{h}0h\bar{l})\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0) {\hbox to 1.05pc{}}(\bar{1}100) {\hbox to 1.1pc{}}(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) {\hbox to 1.05pc{}}(1\bar{1}00) {\hbox to 1.1pc{}}(\bar{1}010)\cr}]
3 b .2. [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Trigonal prism [(11\bar{2}0) \quad(\bar{2}110) \quad(1\bar{2}10)]
        Trigon through origin or [(\bar{1}\bar{1}20)\quad(2\bar{1}\bar{1}0)\quad(\bar{1}2\bar{1}0)]
2 a 3.. [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) {\hbox to 1.05pc{}}(000\bar{1})\cr}]
        Line segment through origin  
1 o 32. [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&2&&1\cr}]  
32 [D_{3}] [Scheme scheme37]  
RHOMBOHEDRAL AXES  
6 c 1 [x,y,z \quad z,x,y\quad y,z,x] Trigonal trapezohedron [\matrix{(hkl) {\hbox to .85pc{}}(lhk) {\hbox to .85pc{}}(klh)\cr}]
      [\bar z,\bar y,\bar x\quad \bar y,\bar x,\bar z\quad \bar x,\bar z,\bar y] Twisted trigonal antiprism [\matrix{(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]
        Ditrigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Truncated trigon through origin [\matrix{(\bar{k}\bar{h}(h\!+\!k)) &(\bar{h}(h\!+\!k)\bar{k}) &((h\!+\!k)\bar{k}\bar{h})\cr}]
        Trigonal dipyramid [\matrix{(hk(2k\!-\!h)) &((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]
        Trigonal prism [\matrix{(\bar{k}\bar{h}(h\!-\!2k)) &(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]
        Rhombohedron [\matrix{(hhl) &(lhh) &(hlh)\cr}]
        Trigonal antiprism [\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]
        Hexagonal prism [\matrix{(11\bar{2})&(\bar{2}11) &(1\bar{2}1)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}2) &(\bar{1}2\bar{1}) &(2\bar{1}\bar{1})\cr}]
3 b .2 [ x,\bar x,0\quad 0,x,\bar x\quad \bar x,0,x] Trigonal prism [(01\bar{1}) \quad(\bar{1}01) \quad(1\bar{1}0)]
        Trigon through origin or [(0\bar{1}1) \quad(10\bar{1}) \quad(\bar{1}10)]
2 a 3. [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o 32 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3m&&2&&1\cr}]  
3m1 [C_{3v}] [Scheme scheme38]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Ditrigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z] Truncated trigon [\matrix{(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
        Ditrigonal prism [\matrix{(hki0)&(ihk0) &(kih0)\cr}]
        Truncated trigon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal pyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Hexagon [\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
3 b .m. [x,\bar x,z \quad x,2x,z\quad 2\bar x,\bar x,z ] Trigonal pyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Trigon  
        Trigonal prism [(10\bar{1}0) \quad(\bar{1}100) \quad(0\bar{1}10)]
        Trigon through origin or [(\bar{1}010) \quad(1\bar{1}00) \quad(01\bar{1}0)]
1 a 3m. [0,0,z ] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&1&&m\cr}]  
31m [C_{3v}] [Scheme scheme39]  
HEXAGONAL AXES  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Ditrigonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z] Truncated trigon [\matrix{(khil) &(hikl) &(ikhl)\cr}]
        Ditrigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Truncated trigon through origin [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal pyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagon [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) &(1\bar{1}00) &(\bar{1}010)\cr}]
3 b ..m [x,0,z\quad 0,x,z \quad \bar x,\bar x,z] Trigonal pyramid [\matrix{(hh\overline{2h}l)&(\overline{2h}hhl)& (h\overline{2h}hl)}]
        Trigon  
        Trigonal prism [(11\bar{2}0) \quad(\bar{2}110) \quad(1\bar{2}10)]
        Trigon through origin or [(\bar{1}\bar{1}20) \quad(2\bar{1}\bar{1}0) \quad(\bar{1}2\bar{1}0)]
1 a 3.m [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox {Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&m&&1\cr}]  
3m [C_{3v}] [Scheme scheme40]  
RHOMBOHEDRAL AXES  
6 c 1 [x,y,z\quad z,x,y\quad y,z,x ] Ditrigonal pyramid [\matrix{(hkl) &(lhk) &(klh)\cr}]
      [z,y,x\quad y,x,z\quad x,z,y ] Truncated trigon [\matrix{(khl) &(hlk) &(lkh)\cr}]
        Ditrigonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h \!+\! k})hk) &(k(\overline{h\! +\! k})h)\cr}]
        Truncated trigon through origin [\matrix{(kh(\overline{h\!+\!k}))&(h(\overline{h \!+\! k})k) &((\overline{h \!+\! k})kh)\cr}]
        Hexagonal pyramid [\matrix{(hk(2k\!-\!h)) &((2k\! -\! h)hk) &(k(2k \!- \!h)h)\cr}]
        Hexagon [\matrix{(kh(2k\!-\!h))&(h(2k\! -\! h)k) &((2k\! - \!h)kh)\cr}]
        Hexagonal prism [\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]
        Hexagon through origin [\matrix{(10\bar{1}) &(0\bar{1}1) &(\bar{1}10)\cr}]
3 b .m [x,y,x \quad x,x,y\quad y,x,x] Trigonal pyramid [\matrix{(hhl) &(lhh) &(hlh)\cr}]
        Trigon  
        Trigonal prism [(11\bar{2}) \quad(\bar{2}11) \quad(1\bar{2}1)]
        Trigon through origin or [(\bar{1}\bar{1}2) \quad(2\bar{1}\bar{1}) \quad(\bar{1}2\bar{1})]
1 a 3m [x,x,x ] Pedion or monohedron [(111) \hbox{ or }(\bar{1}\bar{1}\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 3m&&1&&m\cr}]  
[\openup4pt\matrix{\bar{3}1m\cr \bar{3}1{\displaystyle{2 \over m}}\cr}] [D_{3d}] [Scheme scheme42]  
HEXAGONAL AXES  
12 d 1 [\matrix{x,y,z\quad \bar y,x-y,z \quad \bar x+y,\bar x,z\hfill\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z\hfill}] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkil) &(ihkl) &(kihl)\cr (\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z\hfill}] Trigonal antiprism sliced off by pinacoid [\matrix{(\bar{h}\bar{k}\bar{i}\,\bar{l}) &(\bar{i}\bar{h}\bar{k}\bar{l}) &(\bar{k}\bar{i}\bar{h}\bar{l})\cr (khil) &(hikl)&(ikhl)\cr}]
        Dihexagonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Truncated hexagon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
          [\matrix{(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
          [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagonal prism [\matrix{(0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})\cr}]
          [\matrix{(\bar{h}0h\bar{l}) &(h\bar{h}0\bar{l}) &(0h\bar{h}\bar{l})\cr}]
          [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
6 c ..m [x,0,z \quad 0,x,z \quad \bar x,\bar x,z ] Rhombohedron [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
      [0,\bar x,\bar z \quad \bar x,0,\bar z \quad x,x,\bar z] Trigonal antiprism [\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
6 b ..2 [x,\bar x,0\quad x,2x,0 \quad 2\bar x,\bar x,0] Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
      [ \bar x,x,0 \quad \bar x,2\bar x,0 \quad 2x,x,0 ] Hexagon through origin [\matrix{(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
2 a 3.m [0,0,z \quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3.m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2\cr}]  
[\openup4pt\matrix{\bar{3}m1\hfill\cr \bar{3} {\displaystyle{2 \over m}} 1\hfill\cr}] [D_{3d}] [Scheme scheme41]  
HEXAGONAL AXES  
12 d 1 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\hfill\cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\hfill} ] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkil)\quad (ihkl) \quad(kihl)\cr (khi\bar{l})\quad (hik\bar{l}) \quad(ikh\bar{l})}]
      [\matrix{\bar x,\bar y,\bar z \quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\hfill}] Trigonal antiprism sliced off by pinacoid [\matrix{(\bar{h}\bar{k}\bar{i}\,\bar{l})\quad (\bar{i}\bar{h}\bar{k}\bar{l}) \quad(\bar{k}\bar{i}\bar{h}\bar{l})\cr (\bar{k}\bar{h}\bar{i}l)\quad (\bar{h}\bar{i}\bar{k}l) \quad(\bar{i}\bar{k}\bar{h}l)}]
        Dihexagonal prism [\matrix{(hki0)\quad (ihk0) \quad(kih0)\cr}]
        Truncated hexagon through origin [\matrix{(khi0)\quad (hik0)\quad (ikh0)\cr}]
          [\matrix{(\bar{h}\bar{k}\bar{i}0)\quad (\bar{i}\bar{h}\bar{k}0)\quad(\bar{k}\bar{i}\bar{h}0)\cr}]
          [\matrix{(\bar{k}\bar{h}\bar{i}0)\quad (\bar{h}\bar{i}\bar{k}0) \quad(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal dipyramid [\matrix{(hh\overline{2h}l)\quad (\overline{2h}hhl)\quad (h\overline{2h}hl)\cr}]
        Hexagonal prism [\matrix{(hh\overline{2h}\,\bar{l})\quad (h\overline{2h}h\bar{l})\quad (\overline{2h}hh\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2h\bar{l})\quad(2h\bar{h}\bar{h}\bar{l})\quad (\bar{h}2h\bar{h}\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2hl)\quad (\bar{h}2h\bar{h}l)\quad (2h\bar{h}\bar{h}l)\cr}]
6 c .m. [x,\bar x,z \quad x,2x,z\quad 2\bar x,\bar x,z ] Rhombohedron [\matrix{(h0\bar{h}l)\quad (\bar{h}h0l)\quad (0\bar{h}hl)\cr}]
      [\bar x,x,\bar z \quad 2x,x,\bar z\quad \bar x,2\bar x,\bar z ] Trigonal antiprism [\matrix{(0h\bar{h}\bar{l})\quad (h\bar{h}0\bar{l})\quad (\bar{h}0h\bar{l})\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0)\quad (\bar{1}100)\quad (0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0)\quad (1\bar{1}00)\quad (\bar{1}010)\cr}]
6 b .2. [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Hexagonal prism [\matrix{(11\bar{2}0)\quad (\bar{2}110)\quad (1\bar{2}10)\cr}]
      [\bar x,0,0 \quad 0,\bar x,0 \quad x,x,0 ] Hexagon through origin [\matrix{(\bar{1}\bar{1}20)\quad (\bar{1}2\bar{1}0)\quad (2\bar{1}\bar{1}0)\cr}]
2 a 3m. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001)\quad (000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3 m.] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox {Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2&&2mm\cr}]  
[\openup4pt\matrix{\bar{3}m\hfill\cr \bar{3}{\displaystyle{2 \over m}}\hfill\cr}] [D_{3d}] [Scheme scheme43]  
RHOMBOHEDRAL AXES  
12 d 1 [\matrix{x,y,z\quad z,x,y\quad y,z,x\hfill\cr \bar z,\bar y,\bar x\quad \bar y,\bar x,\bar z\quad \bar x,\bar z,\bar y\hfill }] Ditrigonal scalenohedron or hexagonal scalenohedron [\matrix{(hkl) &(lhk) &(klh)\cr(\bar{k}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{k}) &(\bar{l}\bar{k}\bar{h})\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad \bar z,\bar x,\bar y\quad \bar y,\bar z,\bar x\hfill\cr z,y,x\quad y,x,z\quad x,z,y\hfill }] Trigonal antiprism sliced off by pinacoid [\matrix{(\bar{h}\bar{k}\bar{l}) &(\bar{l}\bar{h}\bar{k}) &(\bar{k}\bar{l}\bar{h})\cr (khl) &(hlk) &(lkh)\cr}]
        Dihexagonal prism [\matrix{(hk(\overline{h\!+\!k})) &((\overline{h\!+\!k})hk) &(k(\overline{h\!+\!k})h)\cr}]
        Truncated hexagon through origin [\matrix{(\bar{k}\bar{h}(h\!+\!k)) &(\bar{h}(h\!+\!k)\bar{k}) &((h\!+\!k)\bar{k}\bar{h})\cr}]
          [\matrix{(\bar{h}\bar{k}(h\!+\!k)) &((h\!+\!k)\bar{h}\bar{k}) &(\bar{k}(h\!+\!k)\bar{h})\cr}]
          [\matrix{(kh(\overline{h\!+\!k})) &(h(\overline{h\!+\!k})k) &((\overline{h\!+\!k})kh)\cr}]
        Hexagonal dipyramid [\matrix{(hk(2k\!-\!h)) &((2k\!-\!h)hk) &(k(2k\!-\!h)h)\cr}]
        Hexagonal prism [\matrix{(\bar{k}\bar{h}(h\!-\!2k)) &(\bar{h}(h\!-\!2k)\bar{k}) &((h\!-\!2k)\bar{k}\bar{h})\cr}]
          [\matrix{(\bar{h}\bar{k}(h\!-\!2k)) &((h\!-\!2k)\bar{h}\bar{k}) &(\bar{k}(h\!-\!2k)\bar{h})\cr}]
          [\matrix{(kh(2k\!-\!h)) &(h(2k\!-\!h)k) &((2k\!-\!h)kh)\cr}]
6 c .m [x,y,x\quad x,x,y\quad y,x,x ] Rhombohedron [\matrix{(hhl) &(lhh) &(hlh)\cr}]
      [\bar x,\bar y,\bar x\quad \bar y,\bar x,\bar x\quad \bar x,\bar x,\bar y ] Trigonal antiprism [\matrix{(\bar{h}\bar{h}\bar{l}) &(\bar{h}\bar{l}\bar{h}) &(\bar{l}\bar{h}\bar{h})\cr}]
        Hexagonal prism [\matrix{(11\bar{2}) &(\bar{2}11) &(1\bar{2}1)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}2) &(\bar{1}2\bar{1}) &(2\bar{1}\bar{1})\cr}]
6 b .2 [x,\bar x,0\quad 0,x,\bar x \quad \bar x,0,x ] Hexagonal prism [\matrix{(01\bar{1}) &(\bar{1}01) &(1\bar{1}0)\cr}]
      [\bar x,x,0\quad 0,\bar x,x \quad x,0,\bar x ] Hexagon through origin [\matrix{(0\bar{1}1) &(10\bar{1}) &(\bar{1}10)\cr}]
2 a 3m [x,x,x\quad \bar x,\bar x,\bar x] Pinacoid or parallelohedron [\matrix{(111) &(\bar{1}\bar{1}\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 3 m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[111]&&\hbox{Along }[1\bar{1}0]&&\hbox{Along }[2\bar{1}\bar{1}]\cr 6mm&&2&&2mm\cr}]  
HEXAGONAL SYSTEM
6 [C_{6}] [Scheme scheme44]  
6 b 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z] Hexagonal pyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z ] Hexagon [\matrix{(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]
        Hexagonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Hexagon through origin [\matrix{(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
1 a 6.. [0,0,z] Pedion or monohedron [(0001) \hbox{ or }(000\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&m&&m}]  
[\bar{6}] [C_{3h}] [Scheme scheme45]  
6 c 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Trigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z] Trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
3 b m.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0 ] Trigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
        Trigon through origin  
2 a 3.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6 ..] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3&&m&&m}]  
[6/m] [C_{6h}] [Scheme scheme46]  
12 c 1 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill \cr Hexagonal\ prism\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr}]
      [\matrix{\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\cr x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z}]   [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l}) \cr(\bar{h}\bar{k}\bar{i}\bar{l}) &(\bar{i}\bar{h}\bar{k}\bar{l}) &(\bar{k}\bar{i}\bar{h}\bar{l})\cr}]
6 b m.. [\matrix{x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0\cr \bar x,\bar y,0\quad y,\bar x+y,0 \quad x-y,x,0 }] [\matrix{\rm Hexagonal\ prism\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr}]
2 a 6.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 6/m.. [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6&&2mm&&2mm\cr}]  
622 [D_{6}] [Scheme scheme47]  
12 d 1 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z \cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z }] [\matrix{{\rm Hexagonal\ trapezohedron}\hfill\cr Twisted\ hexagonal\ antiprism\hfill}] [\matrix{(hkil) &(ihkl)&(kihl) \cr (\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khi\bar{l})&(hik\bar{l}) &(ikh\bar{l}) \cr (\bar{k}\bar{h}\bar{i}\bar{l})&(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
        [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)}]
        [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl) &(h\bar{h}0l) &(0h\bar{h}l)\cr(0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l}) \cr(0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})}]
        [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr(\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr (hh\overline{2h}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l}) \cr(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
6 c ..2 [\matrix{x,\bar x,0\quad x,2x,0\quad 2\bar x,\bar x,0\cr \bar x,x,0\quad \bar x,2\bar x,0 \quad 2x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b .2. [\matrix{x,0,0\quad 0,x,0\quad \bar x,\bar x,0\cr \bar x,0,0\quad 0,\bar x,0\quad x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
2 a 6.. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 622 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2mm\cr}]  
6mm [C_{6v}] [Scheme scheme48]  
12 d 1 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z\cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z }] [\matrix{{\rm Dihexagonal\ pyramid}\hfill\cr Truncated\ hexagon\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khil) &(hikl) &(ikhl) \cr(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
        [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0) &(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
6 c .m. [\matrix{x,\bar x,z\quad x,2x,z\quad 2\bar x,\bar x,z\cr \bar x,x,z\quad \bar x,2\bar x,z\quad 2x,x,z }] [\matrix{{\rm Hexagonal\ pyramid}\hfill\cr Hexagon\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl) &(h\bar{h}0l) &(0h\bar{h}l)\cr}]
        [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b ..m [\matrix{x,0,z\quad 0,x,z\quad \bar x,\bar x,z\cr \bar x,0,z\quad 0,\bar x,z\quad x,x,z }] [\matrix{{\rm Hexagonal\ pyramid}\hfill\cr Hexagon\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr(\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr}]
        [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr(\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
1 a 6mm [0,0,z] Pedion or monohedron [(0001) \hbox{ or } (000\bar{1})]
        Single point  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&m&&m\cr}]  
[\bar{6}m2] [D_{3h}] [Scheme scheme49]  
12 e 1 [\matrix{x,y,z \quad \bar y,x-y,z \quad \bar x+y,\bar x,z\hfill}] Ditrigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [\matrix{x,y,\bar z\quad \bar y,x-y,\bar z \quad \bar x+y,\bar x,\bar z\hfill}] Edge-truncated trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
      [\matrix{\bar y,\bar x,z\quad \bar x+y,y,z \quad x,x-y,z\hfill}]   [\matrix{(\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l)\cr}]
      [\matrix{\bar y,\bar x,\bar z\quad \bar x+y,y,\bar z \quad x,x-y,\bar z\hfill}]   [\matrix{(\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l})&(\bar{i}\bar{k}\bar{h}\bar{l})\cr}]
        Hexagonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
        Hexagonal prism [\matrix{(hh\overline{2h}\,\bar{l}) &(\overline{2h}hh\bar{l}) &(h\overline{2h}h\bar{l})\cr}]
          [\matrix{(\bar{h}\bar{h}2hl) &(\bar{h}2h\bar{h}l) &(2h\bar{h}\bar{h}l)\cr}]
          [\matrix{(\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
6 d m.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0] Ditrigonal prism [\matrix{(hki0) &(ihk0)&(kih0)\cr}]
      [\bar y,\bar x,0\quad \bar x+y,y,0\quad x,x-y,0] Truncated trigon through origin [\matrix{(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)\cr}]
        Hexagonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Hexagon through origin [\matrix{(\bar{1}\bar{1}20) &(\bar{1}2\bar{1}0) &(2\bar{1}\bar{1}0)\cr}]
6 c .m. [x,\bar x,z\quad x,2x,z\quad 2\bar x,\bar x,z] Trigonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
      [x,\bar x,\bar z\quad x,2x,\bar z\quad 2\bar x,\bar x,\bar z ] Trigonal prism [\matrix{(h0\bar{h}\bar{l}) &(\bar{h}h0\bar{l}) &(0\bar{h}h\bar{l})\cr}]
3 b mm2 [x,\bar x,0 \quad x,2x,0 \quad 2\bar x,\bar x,0 ] Trigonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Trigon through origin or [\matrix{(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
2 a 3m. [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6m2] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&m&&2mm\cr}]  
[\bar{6}2m] [D_{3h}] [Scheme scheme50]  
12 e 1 [x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z ] Ditrigonal dipyramid [\matrix{(hkil) &(ihkl) &(kihl)\cr}]
      [x,y,\bar z\quad \bar y,x-y,\bar z\quad \bar x+y,\bar x,\bar z] Edge-truncated trigonal prism [\matrix{(hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr}]
      [y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z]   [\matrix{(khi\bar{l}) &(hik\bar{l}) &(ikh\bar{l})\cr}]
      [y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z]   [\matrix{(khil) &(hikl) &(ikhl)\cr}]
        Hexagonal dipyramid [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl)\cr}]
        Hexagonal prism [\matrix{(h0\bar{h}\bar{l}) &(\bar{h}h0\bar{l}) &(0\bar{h}h\bar{l})\cr}]
          [\matrix{(0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l})\cr}]
          [\matrix{(0h\bar{h}l) &(h\bar{h}0l) &(\bar{h}0hl)\cr}]
6 d m.. [x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0 ] Ditrigonal prism [\matrix{(hki0) &(ihk0) &(kih0)\cr}]
      [y,x,0\quad x-y,\bar y,0\quad \bar x,\bar x+y,0] Truncated trigon through origin [\matrix{(khi0) &(hik0) &(ikh0)\cr}]
        Hexagonal prism [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10)\cr}]
        Hexagon through origin [\matrix{(01\bar{1}0) &(1\bar{1}00) &(\bar{1}010)\cr}]
6 c ..m [x,0,z\quad 0,x,z\quad \bar x,\bar x,z] Trigonal dipyramid [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl)\cr}]
      [x,0,\bar z\quad 0,x,\bar z\quad \bar x,\bar x,\bar z] Trigonal prism [\matrix{(hh\overline{2h}\,\bar{l}) &(\overline{2h}hh\bar{l}) &(h\overline{2h}h\bar{l})\cr}]
3 b m2m [x,0,0\quad 0,x,0\quad \bar x,\bar x,0] Trigonal prism [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10)\cr}]
        Trigon through origin or [(\bar{1}\bar{1}20)\quad (2\bar{1}\bar{1}0)\quad(\bar{1}2\bar{1}0)]
2 a 3.m [0,0,z\quad 0,0,\bar z] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o [\bar 6 2 m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 3m&&2mm&&m\cr}]  
[\openup4pt\!\matrix{6/mmm\hfill\cr \displaystyle{6 \over m}{2 \over m}{2 \over m}\hfill\cr}] [D_{6h}] [Scheme scheme51]  
24 g 1 [\matrix{x,y,z\quad \bar y,x-y,z\quad \bar x+y,\bar x,z\hfill\cr \bar x,\bar y,z\quad y,\bar x+y,z\quad x-y,x,z\hfill \cr y,x,\bar z\quad x-y,\bar y,\bar z\quad \bar x,\bar x+y,\bar z\hfill\cr \bar y,\bar x,\bar z\quad \bar x+y,y,\bar z\quad x,x-y,\bar z\hfill \cr \cr\bar x,\bar y,\bar z\quad y,\bar x+y,\bar z\quad x-y,x,\bar z\hfill\cr x,y,\bar z\quad \bar y,x-y,\bar z \quad \bar x+y,\bar x,\bar z\hfill \cr \bar y,\bar x,z\quad \bar x+y,y,z\quad x,x-y,z\hfill\cr y,x,z\quad x-y,\bar y,z\quad \bar x,\bar x+y,z\hfill \cr}] [\matrix{{\rm Dihexagonal\ dipyramid}\hfill\cr Edge\hbox{-}truncated\ hexagonal\ prism\hfill}] [\matrix{(hkil) &(ihkl) &(kihl) \cr(\bar{h}\bar{k}\bar{i}l) &(\bar{i}\bar{h}\bar{k}l) &(\bar{k}\bar{i}\bar{h}l)\cr (khi\bar{l}) &(hik\bar{l}) &(ikh\bar{l}) \cr(\bar{k}\bar{h}\bar{i}\,\bar{l}) &(\bar{h}\bar{i}\bar{k}\bar{l}) &(\bar{i}\bar{k}\bar{h}\bar{l})\cr\cr (\bar{h}\bar{k}\bar{i}\,\bar{l}) &(\bar{i}\bar{h}\bar{k}\bar{l}) &(\bar{k}\bar{i}\bar{h}\bar{l}) \cr (hki\bar{l}) &(ihk\bar{l}) &(kih\bar{l})\cr (\bar{k}\bar{h}\bar{i}l) &(\bar{h}\bar{i}\bar{k}l) &(\bar{i}\bar{k}\bar{h}l) \cr(khil) &(hikl) &(ikhl)\cr}]
12 f m.. [\matrix{x,y,0\quad \bar y,x-y,0\quad \bar x+y,\bar x,0\cr \bar x,\bar y,0\quad y,\bar x+y,0 \quad x-y,x,0 \cr y,x,0 \quad x-y,\bar y,0\quad \bar x,\bar x+y,0 \cr \bar y,\bar x,0\quad \bar x+y,y,0\quad x,x-y,0}] [\matrix{{\rm Dihexagonal\ prism}\hfill\cr Truncated\ hexagon\ through\ origin\hfill}] [\matrix{(hki0)&(ihk0) &(kih0) \cr(\bar{h}\bar{k}\bar{i}0) &(\bar{i}\bar{h}\bar{k}0) &(\bar{k}\bar{i}\bar{h}0)\cr (khi0) &(hik0) &(ikh0) \cr(\bar{k}\bar{h}\bar{i}0) &(\bar{h}\bar{i}\bar{k}0) &(\bar{i}\bar{k}\bar{h}0)}]
12 e .m. [\matrix{x,2x,z\quad 2\bar x,\bar x,z\quad x,\bar x,z\cr \bar x,2\bar x,z \quad 2x,x,z\quad \bar x,x,z \cr 2x,x,\bar z \quad \bar x,2\bar x,\bar z \quad \bar x,x,\bar z\cr 2\bar x,\bar x,\bar z\quad x,2x,\bar z\quad x,\bar x,\bar z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(h0\bar{h}l) &(\bar{h}h0l) &(0\bar{h}hl) \cr(\bar{h}0hl)&(h\bar{h}0l) &(0h\bar{h}l)\cr (0h\bar{h}\bar{l}) &(h\bar{h}0\bar{l}) &(\bar{h}0h\bar{l}) \cr (0\bar{h}h\bar{l}) &(\bar{h}h0\bar{l}) &(h0\bar{h}\bar{l})\cr}]
12 d ..m [\matrix{x,0,z\quad 0,x,z\quad \bar x,\bar x,z\cr \bar x,0,z\quad 0,\bar x,z \quad x,x,z \cr 0,x,\bar z\quad x,0,\bar z \quad \bar x,\bar x,\bar z \cr 0,\bar x,\bar z \quad \bar x,0,\bar z\quad x,x,\bar z }] [\matrix{{\rm Hexagonal\ dipyramid}\hfill\cr Hexagonal\ prism\hfill}] [\matrix{(hh\overline{2h}l) &(\overline{2h}hhl) &(h\overline{2h}hl) \cr (\bar{h}\bar{h}2hl) &(2h\bar{h}\bar{h}l) &(\bar{h}2h\bar{h}l)\cr (hh\overline{2h}\bar{l}) &(h\overline{2h}h\bar{l}) &(\overline{2h}hh\bar{l}) \cr (\bar{h}\bar{h}2h\bar{l}) &(\bar{h}2h\bar{h}\bar{l}) &(2h\bar{h}\bar{h}\bar{l})\cr}]
6 c mm2 [\matrix{x,2x,0 \quad 2\bar x,\bar x,0\quad x,\bar x,0\cr \bar x,2\bar x,0\quad 2x,x,0 \quad \bar x,x,0 \cr}] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(10\bar{1}0) &(\bar{1}100) &(0\bar{1}10) \cr(\bar{1}010) &(1\bar{1}00) &(01\bar{1}0)\cr}]
6 b m2m [\matrix{x,0,0\quad 0,x,0 \quad \bar x,\bar x,0 \cr \bar x,0,0\quad 0,\bar x,0\quad x,x,0 }] [\matrix{{\rm Hexagonal\ prism}\hfill\cr Hexagon\ through\ origin\hfill}] [\matrix{(11\bar{2}0) &(\bar{2}110) &(1\bar{2}10) \cr (\bar{1}\bar{1}20) &(2\bar{1}\bar{1}0) &(\bar{1}2\bar{1}0)\cr}]
2 a 6mm [0,0,z\quad 0,0,\bar z ] Pinacoid or parallelohedron [\matrix{(0001) &(000\bar{1})\cr}]
        Line segment through origin  
1 o 6mmm [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[100]&&\hbox{Along }[210]\cr 6mm&&2mm&&2mm\cr}]  
CUBIC SYSTEM
23 T [Scheme scheme52]  
12 c 1 [\matrix{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill\cr z,x,y \quad z,\bar x,\bar y\quad \bar z,\bar x,y \quad \bar z,x,\bar y \hfill\cr y,z,x\quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad\bar y,\bar z,x\hfill}] [\matrix{\hbox{Pentagon\hbox{-}tritetrahedron or tetartoid}\hfill\cr \hbox{or tetrahedral pentagon\hbox{-}dodecahedron}\hfill\cr Snub\ tetrahedron\ (= pentagon\hbox{-}tritetra\hbox{-}\hfill\cr hedron + two\ tetrahedra)\hfill\cr}] [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k})\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h)\cr}]
        [\left\{\matrix{\hbox{Trigon-tritetrahedron}\hfill\cr \hbox{or tristetrahedron (for }|h| \,\lt\, |l|\hbox{)}\hfill\cr Tetrahedron\ truncated\ by\ tetrahedron\hfill\cr (for\ |x| \,\lt\, |z|)\hfill\cr \cr \hbox{Tetragon-tritetrahedron or deltohedron}\hfill\cr \hbox{or deltoid-dodecahedron (for}\ |h| \,\gt\, |l|\hbox{)}\hfill\cr Cube\ \&\ two\ tetrahedra\ (for\ |x| \,\gt\, |z|)\hfill\cr}\right\}] [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\hfill\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr}]
        [\matrix{\hbox{Pentagon-dodecahedron}\hfill\cr \hbox{or dihexahedron or pyritohedron}\hfill\cr Irregular\ icosahedron\hfill\cr (=pentagon\hbox{-}dodecahedron + octahedron)\hfill\cr}] [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr(l0k)&(l0\bar{k})&({\bar l}0k)&({\bar l}0\bar{k})\cr(kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0)\cr}]
        [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}] [\openup-1pt\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]
6 b 2.. [\matrix{ x,0,0 \quad \bar x,0,0\hfill\cr 0,x,0\quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\openup-1pt\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
4 a .3. [x,x,x \quad \bar x,\bar x,x\quad \bar x,x,\bar x\quad x,\bar x,\bar x] [\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\hfill\cr}] [\openup-1pt\matrix{{}}(111)&(\bar{1}\bar{1}1)&(\bar{1}1\bar{1})&(1\bar{1}\bar{1})\hfill\cr \hbox{or }(\bar{1}\bar{1}\bar{1})&(11\bar{1})&(1\bar{1}1)&(\bar{1}11)\hfill\cr}\hfill]
1 o 23. [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 2mm&&3&&m\cr}]  
[\openup 6pt\matrix{m{\bar 3}\cr \displaystyle{2 \over m}\bar{3}}] [T_{h}] [Scheme scheme53]  
24 d 1 [\matrix{x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z \hfill\cr z,x,y\quad z,\bar x,\bar y \quad \bar z,\bar x,y\quad \bar z,x,\bar y\hfill\cr y,z,x \quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad \bar y,\bar z,x\hfill\cr \noalign{\vskip10pt}\bar x,\bar y,\bar z \quad x,y,\bar z \quad x,\bar y,z\quad \bar x,y,z \hfill\cr \bar z,\bar x,\bar y\quad \bar z,x,y\quad z,x,\bar y\quad z,\bar x,y\hfill\cr \bar y,\bar z,\bar x \quad y,\bar z,x\quad \bar y,z,x \quad y,z,\bar x\hfill}] [\matrix{\hbox{Didodecahedron or diploid}\hfill\cr\hbox{or dyakisdodecahedron}\hfill\cr Cube\ \&\ octahedron\ \&\hfill\cr pentagon\hbox{-}dodecahedron\hfill\cr}] [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l})\cr(lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k})\cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h)\cr\noalign{\vskip10pt}(\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl)\cr(\bar{l}\bar{h}\bar{k}) &(\bar{l}hk) &(lh\bar{k}) &(l\bar{h}k)\cr (\bar{k}\bar{l}\bar{h}) &(k\bar{l}h) &(\bar{k}lh) &(kl\bar{h})\cr}]
        [\openup-1pt\left\{\matrix{\hbox{Tetragon-trioctahedron or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for}\ |h| \,\lt\, |l|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\ rhomb\hbox{-}\hfill\cr dodecahedron\hfill\cr (for\ |x| \,\lt\, |z|)\hfill\cr \cr \hbox{Trigon-trioctahedron or trisoctahedron}\hfill\cr \hbox{(for }|h| \,\gt\, |l|\hbox{)}\hfill\cr Cube\ truncated\ by\ octahedron\hfill\cr (for\ |x| \,\gt\, |z|)\hfill\cr}\right\}] [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr\noalign{\vskip8pt} (\bar{h}\bar{h}\bar{l}) &(hh\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (\bar{l}\bar{h}\bar{h}) &(\bar{l}hh) &(lh\bar{h}) &(l\bar{h}h)\cr (\bar{h}\bar{l}\bar{h}) &(h\bar{l}h) &(\bar{h}lh) &(hl\bar{h})\cr}]
12 c m.. [\matrix{0,y,z\quad 0,\bar y,z \quad 0,y,\bar z \quad 0,\bar y,\bar z \hfill\cr z,0,y \quad z,0,\bar y \quad \bar z,0,y \quad \bar z,0,\bar y\hfill\cr y,z,0 \quad \bar y,z,0 \quad y,\bar z,0\quad \bar y,\bar z,0\hfill}] [\matrix{\hbox{Pentagon-dodecahedron}\hfill\cr \hbox{or dihexahedron or pyritohedron}\hfill\cr Irregular\ icosahedron\hfill\cr (\!= pentagon\hbox{-}dodecahedron + octahedron)\hfill\cr}] [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l})\cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k})\cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0)\cr}]
        [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}] [\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr(101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr(110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]
8 b .3. [\matrix{ x,x,x\quad \bar x,\bar x,x \quad \bar x,x,\bar x\quad x,\bar x,\bar x\hfill\cr \bar x,\bar x,\bar x \quad x,x,\bar x \quad x,\bar x,x \quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}\hfill] [\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\cr(\bar{1}\bar{1}\bar{1}) &(11\bar{1}) &(1\bar{1}1) &(\bar{1}11)\cr}]
6 a 2mm.. [\matrix{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0 \quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr(010) &(0\bar{1}0)\cr(001) &(00\bar{1})\cr}]
1 o [m\bar3 .] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 2mm&&6&&2mm\cr}]  
432 O [Scheme scheme54]  
24 d 1 [\matrix{x,y,z\quad \bar x,\bar y,z\quad \bar x,y,\bar z\quad x,\bar y,\bar z \hfill\cr z,x,y\quad z,\bar x,\bar y \quad \bar z,\bar x,y\quad \bar z,x,\bar y\hfill\cr y,z,x\quad \bar y,z,\bar x\quad y,\bar z,\bar x\quad \bar y,\bar z,x\hfill\cr \cr y,x,\bar z \quad \bar y,\bar x,\bar z \quad y,\bar x,z \quad \bar y,x,z\hfill\cr x,z,\bar y \quad \bar x,z,y \quad \bar x,\bar z,\bar y\quad x,\bar z,y\hfill\cr z,y,\bar x \quad z,\bar y,x \quad \bar z,y,x \quad \bar z,\bar y,\bar x\hfill}] [\matrix{\hbox{Pentagon-trioctahedron}\hfill\cr \hbox{or gyroid}\hfill\cr \hbox{or pentagon-icositetrahedron}\hfill\cr Snub\ cube\ (=cube\ +\hfill\cr octahedron + pentagon\hbox{-}\hfill\cr trioctahedron)\hfill\cr}] [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) \cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) \cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) \cr\cr (kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr(hl\bar{k}) &(\bar{h}lk) &(\bar{h}\bar{l}\bar{k}) &(h\bar{l}k)\cr (lk\bar{h}) &(l\bar{k}h) &(\bar{l}kh) &(\bar{l}\bar{k}\bar{h})\cr}]
        [\left\{\matrix{\hbox{Tetragon-trioctahedron}\hfill\cr \hbox{or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for }|h| \,\lt\, |l|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\hfill\cr rhomb\hbox{-}dodecahedron\hfill\cr (for\ |x| \,\lt\, |z|)\hfill\cr\cr \hbox{Trigon-trioctahedron}\hfill\cr \hbox{or trisoctahedron}\hfill\cr \hbox{(for }|h| \,\gt\, |l|)\hfill\cr Cube\ truncated\ by\ octahedron\hfill\cr (for\ |x| \,\gt\, |z|)\hfill\cr}\right\}] [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) \cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h}) \cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h) \cr\cr (hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (hl\bar{h}) &(\bar{h}lh) &(\bar{h}\bar{l}\bar{h}) &(h\bar{l}h)\cr (lh\bar{h}) &(l\bar{h}h) &(\bar{l}hh) &(\bar{l}\bar{h}\bar{h})\cr}]
        [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}] [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) \cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) \cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) \cr\cr (k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (0l\bar{k}) &(0lk) &(0\bar{l}\bar{k}) &(0\bar{l}k)\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)\cr}]
12 c ..2 [\matrix{0,y,y \quad 0,\bar y,y \quad 0,y,\bar y \quad 0,\bar y,\bar y \hfill\cr y,0,y \quad y,0,\bar y \quad \bar y,0,y \quad \bar y,0,\bar y\hfill\cr y,y,0 \quad \bar y,y,0 \quad y,\bar y,0 \quad \bar y,\bar y,0\hfill}] [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}] [\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]
8 b .3. [\matrix{x,x,x\quad \bar x,\bar x,x\quad \bar x,x,\bar x \quad x,\bar x,\bar x \hfill\cr x,x,\bar x \quad \bar x,\bar x,\bar x \quad x,\bar x,x \quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}] [\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\cr (11\bar{1}) &(\bar{1}\bar{1}\bar{1})&(1\bar{1}1) &(\bar{1}11)\cr}]
6 a 4.. [\matrix{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0\quad 0,\bar x,0\hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
1 o 432 [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&3m&&2mm\cr}]  
[\bar{4}3m] [T_{d}] [Scheme scheme55]  
24 d 1 [\matrix{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill \cr z,x,y \quad z,\bar x,\bar y \quad \bar z,\bar x,y \quad \bar z,x,\bar y\hfill \cr y,z,x \quad \bar y,z,\bar x\quad y,\bar z,\bar x\quad \bar y,\bar z,x\hfill \cr \cr y,x,z\quad \bar y,\bar x,z \quad y,\bar x,\bar z \quad \bar y,x,\bar z\hfill\cr x,z,y \quad\bar x,z,\bar y \quad\bar x,\bar z,y\quad x,\bar z, \bar y\hfill \cr z,y,x \quad z,\bar y,\bar x \quad \bar z,y,\bar x\quad \bar z,\bar y,x\hfill}] [\matrix{\hbox{Hexatetrahedron}\hfill\cr \hbox{or hexakistetrahedron}\hfill\cr Cube\ truncated\ by\hfill\cr two\ tetrahedra\hfill\cr}] [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) \cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) \cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) \cr\cr (khl) &(\bar{k}\bar{h}l) &(k\bar{h}\bar{l}) &(\bar{k}h\bar{l})\cr (hlk) &(\bar{h}l\bar{k}) &(\bar{h}\bar{l}k) &(h\bar{l}\bar{k})\cr (lkh) &(l\bar{k}\bar{h}) &(\bar{l}k\bar{h}) &(\bar{l}\bar{k}h)\cr}]
        [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\ by\ cube\hfill\cr}] [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) \cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) \cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) \cr\cr (k0l) &(\bar{k}0l) &(k0\bar{l}) &(\bar{k}0\bar{l})\cr (0lk) &(0l\bar{k}) &(0\bar{l}k) &(0\bar{l}\bar{k})\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)}]
12 c ..m [\matrix{x,x,z \quad \bar x,\bar x,z \quad \bar x,x,\bar z \quad x,\bar x,\bar z \hfill\cr z,x,x \quad z,\bar x,\bar x \quad \bar z,\bar x,x \quad \bar z,x,\bar x\hfill\cr x,z,x \quad \bar x,z,\bar x \quad x,\bar z,\bar x \quad \bar x,\bar z,x\hfill}] [\left\{\matrix{\hbox{Trigon-tritetrahedron}\hfill\cr \hbox{or tristetrahedron}\hfill\cr \hbox{(for }|h| \,\lt\, |l|\hbox{)}\hfill\cr Tetrahedron\ truncated\hfill\cr by\ tetrahedron\hfill\cr (for\ |x| \,\lt \,|z|)\hfill\cr \cr\hbox{Tetragon-tritetrahedron}\hfill\cr \hbox{or deltohedron}\hfill\cr \hbox{or deltoid-dodecahedron}\hfill\cr \hbox{(for }|h|\,\gt\, |l|\hbox{)}\hfill\cr Cube\ \&\ two\ tetrahedra\hfill\cr (for\ |x| \,\gt\, |z|)\hfill\cr}\right\}] [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l})\cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h})\cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h)\cr}]
        [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}] [\matrix{(110) &(\bar{1}\bar{1}0) &(\bar{1}10) &(1\bar{1}0)\cr (011) &(0\bar{1}\bar{1}) &(0\bar{1}1) &(01\bar{1})\cr (101) &(\bar{1}0\bar{1}) &(10\bar{1}) &(\bar{1}01)\cr}]
6 b 2.mm [\matrix{x,0,0 \quad \bar x,0,0 \hfill\cr 0,x,0 \quad 0,\bar x,0 \hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
4 a .3m [x,x,x \quad \bar x,\bar x,x\quad \bar x,x,\bar x\quad x,\bar x,\bar x] [\matrix{\hbox{Tetrahedron}\hfill\cr Tetrahedron\hfill\cr}] [\matrix{}(111) {\hbox to .65pc{}}(\bar{1}\bar{1}1) {\hbox to .65pc{}}(\bar{1}1\bar{1}) {\hbox to .7pc{}}(1\bar{1}\bar{1})\cr \hbox{or }(\bar{1}\bar{1}\bar{1}) {\hbox to .65pc{}}(11\bar{1}) {\hbox to .65pc{}}(1\bar{1}1) {\hbox to .75pc{}}(\bar{1}11)\cr}]
1 o [\bar 4 3 m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&3m&&m\cr}]  
[\openup 6pt\matrix{m\bar{3}m\cr\displaystyle{4 \over m}\bar{3}{2 \over m}\cr}] [O_{h}] [Scheme scheme56]  
48 f 1 [\matrix{x,y,z \quad \bar x,\bar y,z \quad \bar x,y,\bar z \quad x,\bar y,\bar z \hfill\cr z,x,y \quad z,\bar x,\bar y \quad \bar z,\bar x,y \quad \bar z,x,\bar y\hfill\cr y,z,x \quad \bar y,z,\bar x \quad y,\bar z,\bar x \quad \bar y,\bar z,x\hfill\cr\cr y,x,\bar z \quad \bar y,\bar x,\bar z \quad y,\bar x,z \quad\bar y,x,z \hfill\cr x,z,\bar y \quad \bar x,z,y \quad \bar x,\bar z,\bar y \quad x,\bar z,y\hfill\cr z,y,\bar x \quad z,\bar y,x \quad \bar z,y,x \quad\bar z,\bar y,\bar x\hfill\cr \cr \bar x,\bar y,\bar z \quad x,y,\bar z \quad x,\bar y,z \quad \bar x,y,z\hfill\cr \bar z,\bar x,\bar y \quad \bar z,x,y \quad z,x,\bar y \quad z,\bar x,y\hfill\cr \bar y,\bar z,\bar x \quad y,\bar z,x \quad \bar y,z,x \quad y,z,\bar x\hfill\cr\cr \bar y,\bar x,z \quad y,x,z \quad \bar y,x,\bar z \quad y,\bar x,\bar z\hfill\cr \bar x,\bar z,y \quad x,\bar z,\bar y \quad x,z,y \quad \bar x,z,\bar y\hfill\cr \bar z,\bar y,x \quad \bar z,y,\bar x \quad z,\bar y,\bar x \quad z,y,x\hfill}] [\matrix{\hbox{Hexaoctahedron}\hfill\cr \hbox{or hexakisoctahedron}\hfill\cr Cube\ truncated\ by\hfill\cr octahedron\ and\ by\ rhomb\hbox{-}\hfill\cr dodecahedron\hfill\cr}] [\matrix{(hkl) &(\bar{h}\bar{k}l) &(\bar{h}k\bar{l}) &(h\bar{k}\bar{l}) \cr (lhk) &(l\bar{h}\bar{k}) &(\bar{l}\bar{h}k) &(\bar{l}h\bar{k}) \cr (klh) &(\bar{k}l\bar{h}) &(k\bar{l}\bar{h}) &(\bar{k}\bar{l}h) \cr\cr (kh\bar{l}) &(\bar{k}\bar{h}\bar{l}) &(k\bar{h}l) &(\bar{k}hl)\cr (hl\bar{k}) &(\bar{h}lk) &(\bar{h}\bar{l}\bar{k}) &(h\bar{l}k)\cr (lk\bar{h}) &(l\bar{k}h) &(\bar{l}kh) &(\bar{l}\bar{k}\bar{h})\cr\cr (\bar{h}\bar{k}\bar{l}) &(hk\bar{l}) &(h\bar{k}l) &(\bar{h}kl) \cr (\bar{l}\bar{h}\bar{k}) &(\bar{l}hk) &(lh\bar{k}) &(l\bar{h}k) \cr (\bar{k}\bar{l}\bar{h}) &(k\bar{l}h) &(\bar{k}lh) &(kl\bar{h}) \cr\cr (\bar{k}\bar{h}l) &(khl) &(\bar{k}h\bar{l}) &(k\bar{h}\bar{l})\cr (\bar{h}\bar{l}k) &(h\bar{l}\bar{k}) &(hlk) &(\bar{h}l\bar{k})\cr (\bar{l}\bar{k}h) &(\bar{l}k\bar{h}) &(l\bar{k}\bar{h}) &(lkh)\cr}]
24 e ..m [\matrix{ x,x,z \quad \bar x,\bar x,z \quad \bar x,x,\bar z \quad x,\bar x,\bar z\hfill\cr z,x,x \quad z,\bar x,\bar x \quad \bar z,\bar x,x \quad \bar z,x,\bar x\hfill\cr x,z,x \quad \bar x,z,\bar x \quad x,\bar z,\bar x \quad \bar x,\bar z,x\hfill\cr\cr x,x,\bar z \quad \bar x,\bar x,\bar z \quad x,\bar x,z \quad \bar x,x,z \hfill \cr x,z,\bar x \quad \bar x,z,x \quad \bar x,\bar z,\bar x \quad x,\bar z,x\hfill\cr z,x,\bar x \quad z,\bar x,x\quad \bar z,x,x \quad \bar z,\bar x,\bar x\hfill}] [\left\{\matrix{\hbox{Tetragon-trioctahedron}\hfill\cr \hbox{or trapezohedron}\hfill\cr \hbox{or deltoid-icositetrahedron}\hfill\cr \hbox{(for }|h| \,\lt \,|l|\hbox{)}\hfill\cr Cube\ \&\ octahedron\ \&\ rhomb\hbox{-}\hfill\cr dodecahedron\hfill\cr (for\ |x| \,\lt\, |z|)\hfill\cr\cr \hbox{Trigon-trioctahedron}\hfill\cr \hbox{or trisoctahedron}\hfill\cr \hbox{(for }|h| \,\gt\, |l|)\hfill\cr Cube\ truncated\ by\hfill\cr octahedron\hfill\cr (for\ |x| \,\gt\, |z|)\hfill\cr}\right\}] [\matrix{(hhl) &(\bar{h}\bar{h}l) &(\bar{h}h\bar{l}) &(h\bar{h}\bar{l}) \cr (lhh) &(l\bar{h}\bar{h}) &(\bar{l}\bar{h}h) &(\bar{l}h\bar{h}) \cr (hlh) &(\bar{h}l\bar{h}) &(h\bar{l}\bar{h}) &(\bar{h}\bar{l}h) \cr\cr (hh\bar{l}) &(\bar{h}\bar{h}\bar{l}) &(h\bar{h}l) &(\bar{h}hl)\cr (hl\bar{h}) &(\bar{h}lh) &(\bar{h}\bar{l}\bar{h}) &(h\bar{l}h)\cr (lh\bar{h}) &(l\bar{h}h) &(\bar{l}hh) &(\bar{l}\bar{h}\bar{h})\cr}]
24 d m.. [\matrix{ 0,y,z \quad 0,\bar y,z \quad 0,y,\bar z \quad 0,\bar y,\bar z\hfill\cr z,0,y \quad z,0,\bar y \quad \bar z,0,y \quad \bar z,0,\bar y\hfill\cr y,z,0 \quad \bar y,z,0 \quad y,\bar z,0 \quad \bar y,\bar z,0\hfill\cr\cr y,0,\bar z \quad \bar y,0,\bar z \quad y,0,z \quad \bar y,0,z \hfill\cr 0,z,\bar y \quad 0,z,y \quad 0,\bar z,\bar y \quad 0,\bar z,y\hfill\cr z,y,0 \quad z,\bar y,0 \quad \bar z,y,0 \quad \bar z,\bar y,0\hfill}] [\matrix{\hbox{Tetrahexahedron}\hfill\cr \hbox{or tetrakishexahedron}\hfill\cr Octahedron\ truncated\hfill\cr by\ cube\hfill\cr}] [\matrix{(0kl) &(0\bar{k}l) &(0k\bar{l}) &(0\bar{k}\bar{l}) \cr (l0k) &(l0\bar{k}) &(\bar{l}0k) &(\bar{l}0\bar{k}) \cr (kl0) &(\bar{k}l0) &(k\bar{l}0) &(\bar{k}\bar{l}0) \cr\cr (k0\bar{l}) &(\bar{k}0\bar{l}) &(k0l) &(\bar{k}0l)\cr (0l\bar{k}) &(0lk) &(0\bar{l}\bar{k}) &(0\bar{l}k)\cr (lk0) &(l\bar{k}0) &(\bar{l}k0) &(\bar{l}\bar{k}0)\cr}]
12 c m.m2 [\matrix{0,y,y \quad 0,\bar y,y \quad 0,y,\bar y \quad 0,\bar y,\bar y\hfill\cr y,0,y \quad y,0,\bar y \quad \bar y,0,y \quad \bar y,0,\bar y\hfill\cr y,y,0 \quad \bar y,y,0 \quad y,\bar y,0 \quad \bar y,\bar y,0\hfill}] [\matrix{\hbox{Rhomb-dodecahedron}\hfill\cr Cuboctahedron\hfill\cr}] [\matrix{(011) &(0\bar{1}1) &(01\bar{1}) &(0\bar{1}\bar{1})\cr (101) &(10\bar{1}) &(\bar{1}01) &(\bar{1}0\bar{1})\cr (110) &(\bar{1}10) &(1\bar{1}0) &(\bar{1}\bar{1}0)\cr}]
8 b .3m [\matrix{x,x,x\quad \bar x,\bar x,x \quad \bar x,x,\bar x \quad x,\bar x,\bar x \hfill\cr x,x,\bar x \quad \bar x,\bar x,\bar x \quad x,\bar x,x\quad \bar x,x,x\hfill}] [\matrix{\hbox{Octahedron}\hfill\cr Cube\hfill\cr}] [\matrix{(111) &(\bar{1}\bar{1}1) &(\bar{1}1\bar{1}) &(1\bar{1}\bar{1})\hfill\cr (11\bar{1}) &(\bar{1}\bar{1}\bar{1}) &(1\bar{1}1) &(\bar{1}11)\hfill\cr}]
6 a 4m.m [\matrix{x,0,0 \quad \bar x,0,0\hfill\cr 0,x,0 \quad 0,\bar x,0 \hfill\cr 0,0,x \quad 0,0,\bar x\hfill}] [\matrix{\hbox{Cube or hexahedron}\hfill\cr Octahedron\hfill\cr}] [\matrix{(100) &(\bar{1}00)\cr (010) &(0\bar{1}0)\cr (001) &(00\bar{1})\cr}]
1 o [m\bar3m] [0,0,0] Point in origin  
        Symmetry of special projections  
        [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[110]\cr 4mm&&6mm&&2mm\cr}]