Input interpretation
osmium | Bravais lattice
Result
simple hexagonal
Image
Image
Unit cell relations
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)
Lattice invariants
dimension | 3 determinant | 3
Lattice-packing invariants
volume | sqrt(3)≈1.73205
Quadratic form and theta series
quadratic form | x^2 + 2 y^2 + 2 y z + 2 z^2
More properties
number of symmetries | 24
Common properties
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
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
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