Basic Examples 
(2)
 

Look up a single group. p4m, the symmetry group of a square grid with mirrors and diagonals, has
𝐷
4
point symmetry and four mirror axes plus two glide axes per fundamental domain:
In[1]:=
wallpaperGroups["p4m"]
Out[1]=
FullSymbolp4mm,Orbifold*442,LatticeSquare,PointGroupD4,RotationOrders{4,4,2},Mirrors4,Glides2
The dataset is a flat association of 17 entries:
In[2]:=
Length[wallpaperGroups]
Out[2]=
17