I have very little experience and
knowledge about groups theory, even though I studied it in 1972.
I read from some books about the 17 wall-paper groups.
I shall mention the following famous ones:
* Hilbert Cohn-Vossen Geometry and Imagination
* Grossman Magnus The groups and their graphs
* Weyl The simmetry
* Coxeter Introduction to geometry
I'm thinking to the very celebrated hollandisch painter M.C. Escher,
mentioned by Coxeter in his previously cited work.
According to Coxeter's notation these groups are represented by these
symbols with the following meaning:
p1 two translations
p2 three half turns
pg two parallel glide reflections
pm two reflections and a translation
cm a reflection and a parallel glide reflection
pmm reflection in the four sides of a rectangle
pmg A reflection and two half-turns
pgg Two perpendicular glide reflections
cmm two perpendicular reflection and a half-turn
pgg two perpendicular reflection and a half-turn
p4 a half turn and a quarter turn
p4m Reflections in three sides of a (45,45,90) triangle
p4g A reflection and a quarter-turn
p3 Two rotations through 120
p3m1 A reflection and a rotation through 120
p31m Reflection in the three sides of an equilateral triangle
p6 A half-turn and a rotation through 120
p6m Reflections in the three sides of a (30,60,90) triangle
According Grossman Magnus's book the wall-paper figures are
present in the graphs which can completely cover the plane
by a fundamental region.
I have little experience with GAP too.
I know there's a GAP-package for crystallographic groups.
Now my question:
How can I identify or build the 17 groups by GAP?
Thank you very much in advance.