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Dear gap-forum,

I am sorry, there were two wrong lines in my first answer

to Giancarlo Bassi's letter. Here is a new attempt.

Giancarlo Bassi wrote:

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?

---------------------------------------------------------------------

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> DisplaySpaceGroupType( "p2gg" ); #I Space-group type (2,2,2,1,3); IT(8) = p2gg; orbit size 1

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)

Miles-Receive-Header: reply

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