Template:Frieze group notations

Frieze groups

IUC	Cox.	Schön.*	Orbifold	Diagram§	Examples and Conway nickname		Description
p1	[∞]+	C∞ Z∞	∞∞			hop	(T) Translations only: This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g	[∞+,2+]	S∞ Z∞	∞×			step	(TG) Glide-reflections and Translations: This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
p1m1	[∞]	C∞v Dih∞	*∞∞			sidle	(TV) Vertical reflection lines and Translations: The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis.
p2	[∞,2]+	D∞ Dih∞	22∞			spinning hop	(TR) Translations and 180° Rotations: The group is generated by a translation and a 180° rotation.
p2mg	[∞,2+]	D∞d Dih∞	2*∞			spinning sidle	(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° Rotations: The translations here arise from the glide reflections, so this group is generated by a glide reflection and either a rotation or a vertical reflection.
p11m	[∞+,2]	C∞h Z∞×Dih1	∞*			jump	(THG) Translations, Horizontal reflections, Glide reflections: This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
p2mm	[∞,2]	D∞h Dih∞×Dih1	*22∞			spinning jump	(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations: This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.

^*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries

^§The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.