Input interpretation
![thallium | Bravais lattice](../image_source/58d3b7523f65fe8c1379b0d920f0a6c5.png)
thallium | Bravais lattice
Result
![simple hexagonal](../image_source/96cb73dd0a2f9279a0ca99302fa6054b.png)
simple hexagonal
Image
![Image](../image_source/a9f33e938c5bdb807097699c9f9181ec.png)
Image
Unit cell relations
![angle relations | α = 90°, γ = 120° edge relations | a!=c unit cell volume | 1/2 sqrt(3) a^2 c](../image_source/dee8c50f29d36f073cd4c4e0bef88a12.png)
angle relations | α = 90°, γ = 120° edge relations | a!=c unit cell volume | 1/2 sqrt(3) a^2 c
Description of lattice
![Gram matrix | (1 | 0 | 0 0 | 2 | 1 0 | 1 | 2)](../image_source/4c3b67b6bddfc2e846a9060a9a136829.png)
Gram matrix | (1 | 0 | 0 0 | 2 | 1 0 | 1 | 2)
Lattice invariants
![dimension | 3 determinant | 3](../image_source/bd9f9a91e0d486594c7b544215aa1b76.png)
dimension | 3 determinant | 3
Lattice-packing invariants
![volume | sqrt(3)≈1.73205](../image_source/56e66149cd9ec32cbb08261d7ec04745.png)
volume | sqrt(3)≈1.73205
Quadratic form and theta series
![quadratic form | x^2 + 2 y^2 + 2 y z + 2 z^2](../image_source/74c5bafb5ef769a6de9126e129c054e4.png)
quadratic form | x^2 + 2 y^2 + 2 y z + 2 z^2
More properties
![number of symmetries | 24](../image_source/6b0c19bcd60adc6c6197e7040828cf8e.png)
number of symmetries | 24
Common properties
![even | integral | nonunimodular](../image_source/11ea36d30aad85824c704e2a15b738e4.png)
even | integral | nonunimodular
Crystallographic properties
![lattice system | hexagonal crystal system | trigonal | hexagonal crystal family | hexagonal required point group symmetry | 1 3-fold rotation axis | 1 6-fold rotation axis point groups | 12 space groups | 45](../image_source/23833d6368d66981eb577ff6e518f2e1.png)
lattice system | hexagonal crystal system | trigonal | hexagonal crystal family | hexagonal required point group symmetry | 1 3-fold rotation axis | 1 6-fold rotation axis point groups | 12 space groups | 45
Point groups
![crystal class | Schönflies | Hermann-Mauguin trigonal pyramidal | C_3 | 3 rhombohedral | S_6 | 3^_ trigonal trapezoidal | D_3 | 32 ditrigonal pyramidal | C_3v | 3m ditrigonal scalahedral | D_3d | 3^_m hexagonal pyramidal | C_6 | 6 trigonal dipyramidal | C_3h | 6^_ hexagonal dipyramidal | C_6h | 6/m hexagonal trapezoidal | D_6 | 622 dihexagonal pyramidal | C_6v | 6mm ditrigonal dipyramidal | D_3h | 6^_m2 dihexagonal dipyramidal | D_6h | 6/mmm](../image_source/7ad745e9ebb192e142f1e1a8877a1a57.png)
crystal class | Schönflies | Hermann-Mauguin trigonal pyramidal | C_3 | 3 rhombohedral | S_6 | 3^_ trigonal trapezoidal | D_3 | 32 ditrigonal pyramidal | C_3v | 3m ditrigonal scalahedral | D_3d | 3^_m hexagonal pyramidal | C_6 | 6 trigonal dipyramidal | C_3h | 6^_ hexagonal dipyramidal | C_6h | 6/m hexagonal trapezoidal | D_6 | 622 dihexagonal pyramidal | C_6v | 6mm ditrigonal dipyramidal | D_3h | 6^_m2 dihexagonal dipyramidal | D_6h | 6/mmm
Space groups
![crystal class | IUCr number | Hermann-Mauguin trigonal pyramidal | 143 | 144 | 145 | P3 | P31 | P32 rhombohedral | 147 | P3^_ trigonal trapezoidal | 149 | 150 | 151 | 152 | 153 | 154 | P312 | P321 | P3112 | P3121 | P3212 | P3221 ditrigonal pyramidal | 156 | 157 | 158 | 159 | P3m1 | P31m | P3c1 | P31c ditrigonal scalahedral | 162 | 163 | 164 | 165 | P31m^_ | P31c^_ | P3m1^_ | P3c1^_ hexagonal pyramidal | 168 | 169 | 170 | 171 | 172 | 173 | P6 | P61 | P65 | P62 | P64 | P63 trigonal dipyramidal | 174 | P6^_ hexagonal dipyramidal | 175 | 176 | P6/m | P63/m hexagonal trapezoidal | 177 | 178 | 179 | 180 | 181 | 182 | P622 | P6122 | P6522 | P6222 | P6422 | P6322 dihexagonal pyramidal | 183 | 184 | 185 | 186 | P6mm | P6cc | P63cm | P63mc ditrigonal dipyramidal | 187 | 188 | 189 | 190 | P6m2^_ | P6c2^_ | P62m^_ | P62c^_ dihexagonal dipyramidal | 191 | 192 | 193 | 194 | P6/mmm | P6/mcc | P63/mcm | P63/mmc](../image_source/2e3dc4bc3cfc9a450144fdc5c5ebaa68.png)
crystal class | IUCr number | Hermann-Mauguin trigonal pyramidal | 143 | 144 | 145 | P3 | P31 | P32 rhombohedral | 147 | P3^_ trigonal trapezoidal | 149 | 150 | 151 | 152 | 153 | 154 | P312 | P321 | P3112 | P3121 | P3212 | P3221 ditrigonal pyramidal | 156 | 157 | 158 | 159 | P3m1 | P31m | P3c1 | P31c ditrigonal scalahedral | 162 | 163 | 164 | 165 | P31m^_ | P31c^_ | P3m1^_ | P3c1^_ hexagonal pyramidal | 168 | 169 | 170 | 171 | 172 | 173 | P6 | P61 | P65 | P62 | P64 | P63 trigonal dipyramidal | 174 | P6^_ hexagonal dipyramidal | 175 | 176 | P6/m | P63/m hexagonal trapezoidal | 177 | 178 | 179 | 180 | 181 | 182 | P622 | P6122 | P6522 | P6222 | P6422 | P6322 dihexagonal pyramidal | 183 | 184 | 185 | 186 | P6mm | P6cc | P63cm | P63mc ditrigonal dipyramidal | 187 | 188 | 189 | 190 | P6m2^_ | P6c2^_ | P62m^_ | P62c^_ dihexagonal dipyramidal | 191 | 192 | 193 | 194 | P6/mmm | P6/mcc | P63/mcm | P63/mmc