VW  CVM 1.1 
These five groups are intimately related, so we handle them all in a single table.
The possibilities for homomorphisms from these groups is smaller than one might guess from the list of generators, because t_{1} and R_{3} can never be negating isometries of any pattern. To see this note that R cannot go to 1 because its order is odd, and that t_{1} cannot go to 1, because then t_{2} and the product t_{2} t_{1} would also be taken to 1, which would give a contradiction.
We group all the homomorphisms in one table.
type  P(k) = 1  wallpaper type of kernel group, with remarks 
p3  none  E = G = p3 
p31m  none  E = G = p31m 
p31m'  M_{x}  E = p31m, G = p3 = {t_{1}, R_{3}} 
p3m1  none  E = G = p3m1 
p3m'  M_{y}  E = p3m1, G = p3 
p6  none  E = G = p6 
p6'  R_{6}  E = p6,

p6m  none  E = G = p6m 
p6m'm'  M_{x}  E = p6m, G = p6 
p6'm'm  R_{6}  E = p6m,

p6'mm'  R_{6} and M_{x}  E = p6m,


