Giancarlo Bassi wrote to the gap-forum:

I read from some books about the 17 wall-paper groups.

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?

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Dear Giancarlo Bassi,

the two-dimensional wall-paper groups are available in GAP under the
following Hermann-Mauguin symbols:

"p1", "p2", "pm", "pg", "cm", "p2mm", "p2mg", "p2gg", "c2mm", "p4",
"p4mm", "p4gm", "p3", "p3m1", "p31m", "p6", "p6mm".

Note that some of these symbols differ from those in your list.

GAP provides display commands like

gap> DisplaySpaceGroupGenerators( "p2gg" );
#I Non-translation generators of SpaceGroupOnLeftBBNWZ( 2, 2, 2, 1, 3 )

or

gap> DisplaySpaceGroupGenerators( "p2gg" );
#I Non-translation generators of SpaceGroupOnLeftBBNWZ( 2, 2, 2, 1, 3 )

[ [    1,    0,  1/2 ],
  [    0,   -1,  1/2 ],
  [    0,    0,    1 ] ]

[ [  -1,   0,   0 ],
  [   0,  -1,   0 ],
  [   0,   0,   1 ] ]

or commands to actually construct the groups like

gap> s := SpaceGroupBBNWZ( "p2gg" );
SpaceGroupOnRightBBNWZ( 2, 2, 2, 1, 3 )

You should perhaps look through the CrystCat manual to get the full
list of the GAP functions that accept an Hermann-Mauguin symbol as
argument.

With kind regards, Volkmar Felsch (Aachen)

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