The pictures on this page were borrowed from
http://www.clarku.edu/~djoyce/wallpaper/seventeen.html without
permission. I do hope I will not be sued.
This group contains only translations. The lattice is a
"parallelogram".
G2: translations and order 2 rotation. The lattice is a
"parallelogram".
G3: translations and rotation of order 3. The lattice is a "hexagon".
G4: translations and order 4 rotation. The lattice is a "square".
G6: translations and order 6 rotation (hence also contains rotations
of order 2 and 3). The lattice is a "hexagon".
G11: translation and reflection with axis parallel to one of the
translations; there are no rotations. The lattice is a "rectangle".
G12: translations and reflection with axis parallel to that of a
translation. The axes of translations can intersect at any angle and
the axis of a reflection bisects that angle. The lattice is a "rhombus".
G13: translations and a glide reflection with the axis parallel to one
translation and perpendicular to the other. There are no
reflections. The lattice is a "rectangle".
G21: translations, order 2 rotation, and reflection with axis parallel
to one of the translations. The lattice is a "square".
G22: translations, order 2 rotation, reflection with axis parallel to
one translation whose product with a rotation gives a glide reflection
parallel to the other translation.
G23: translations, rotation of order two, reflection with axis
bisecting the angle between the axes of translations and reflection
perpendicular to it. The lattice is a "rhombus".
G24: translations, rotation of order 2, glide reflection parallel to
one of the translations and a glide reflection parallel to the other
translation. There are no reflections. The lattice is a "rectangle".
G41: translations, order 4 rotation (and order 2) and reflection
parallel to one of the translations. The lattice is a "square".
G42: translations, order 4 rotations, glide reflection parallel to one
of the translations (and reflection perpendicular to it). The lattice
is a "square".
G31: translations, order 3 rotation, reflection parallel to one
translation (all the reflection axes make 60 degrees with each
other). The lattice is a "hexagon".
G32: translations, order 3 rotation, reflection with axis bisecting
the angle of the translations (and all others have axes making 60
degrees with each other). The lattice is hexagonal.
G61: translations, order 6 rotations (and order 2 and 3),
reflections. The lattice is hexagonal.
