structure of materials point groups

cubic crystal

a = b = c

all angles = 90

lattice: cubic

symmetry: 3-fold along diagonals

point groups: 23, m-3, -43m, 432, m-3m

tetragonal crystal

a = b != c

all angles = 90

lattice: tetragonal

symmetry: 4-fold axis

point groups: 4, -4, 4/m, 422, 4mm, -4m2, 4/mmm

hexagonal crystal

a = b != c

alpha = beta = 90, gamma = 120

lattice: hexagonal

symmetry: 6-fold axis

point groups: 6, -6, 6/m, 622, 6mm, -6m2, 6/mmm

trigonal crystal

lattice: hexagonal (a = b != c, alpha = beta = 90, gamma =120) or rhombohedral (a = b = c, angles != 90)

symmetry: 3-fold axis

point groups: 3, -3, 32, 3m, -3m

orthorhombic crystal

a != b != c

all angles = 90

lattice: orthorhombic

symmetry: 3 perpendicular 2-fold axes or mirror planes

point groups: 222, 2mm, mmmm

monoclinic crystal

a != b != c

alpha = gamma = 90, beta != 90

lattice: monoclinic

symmetry: 2-fold or mirror plane

point groups: 2, m, 2/m

triclinic crystal

a != b != c

no angles equal each other

lattice: triclinic

symmetry: non

point groups: 1, -1