VW - CVM 1.1

The Groups p3, p31m, p3m1, p6, and p6m

These five groups are intimately related, so we handle them all in a single table.

	Generators:		p3  =  {t1, R3} p31m = {t1, R3, Mx} p3m1 = {t1, R3, My} p6 = {t1, R6} p6m = {t1, R6, Mx}
	Relations:		t2 = R3 t1 R3-1, R33 = e, R3-1 t1 R3 = t1-1 t2-1 R62 = R3, R6 Mx = My
	Cell diagram:

The possibilities for homomorphisms from these groups is smaller than one might guess from the list of generators, because t₁ and R₃ 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 t₁ cannot go to -1, because then t₂ and the product t₂ t₁ would also be taken to -1, which would give a contradiction.

We group all the homomorphisms in one table.

Table: Eleven homomorphisms from groups related to p3

type	P(k) = -1	wallpaper type of kernel group, with remarks
p3	none	E = G = p3
p31m	none	E = G = p31m
p31m'	Mx	E = p31m, G = p3 = {t1, R3}
p3m1	none	E = G = p3m1
p3m'	My	E = p3m1, G = p3
p6	none	E = G = p6
p6'	R6	E = p6, G = p3 = {t1, R62}
p6m	none	E = G = p6m
p6m'm'	Mx	E = p6m, G = p6
p6'm'm	R6	E = p6m, G = p31m = {t1, R62, Mx}; My negative
p6'mm'	R6 and Mx	E = p6m, G = p3m1 = {t1, R62, R6 Mx}; My positive

Ways to Visualize Wallpaper Functions Why there are 63 Types The Algebra of Wallpapers The Group p4m