Factorization

Generalized translation and axial point group: G = T̂ ∘ P

Generalized translational group T̂: product of two of the following 4 infinite cyclic groups

2D Generalized translational groups T̂: product of two 1D generalized translational groups

T̂ T₁T₂ LG

T T(a) ⊗T(b) 1,2,3,4,6,8,10,11,13,14,18,19,22,23,26,27,35,37,47,49,50,51,53,55,57,59,61,65--80

T_h T(a) ⊗ T_h(b) 5,7,31,38,41

T(a) ⊗T_h(½(a+b)) 2,34,36,39,42,46,48,52,62,64

T_v T(a) ∧ T_v(b) 12,16,24,30,40

T(a) ∧T_v(½(a+b)) 56,58,63

T_U T(a) ∧ T_U(b) 9,15,20,28,29

T(a) ∧T_U(½(a+b)) 54,60

T_Uh T_U(a) ∘T_h(b) 33,43,45

T_Uv T_U(a) ∘ T_v(b) 17

T_UU T_U(a) ∘ T_U(b) 21

T_vv T_v(a) ∘ T_v(b) 25,44

Axial point group P: one of the groups from the following 7 families for n=1,2,3,4,6

T̂	T₁T₂	LG
T	T(a) ⊗T(b)	1,2,3,4,6,8,10,11,13,14,18,19,22,23,26,27,35,37,47,49,50,51,53,55,57,59,61,65--80
T_h	T(a) ⊗ T_h(b)	5,7,31,38,41
	T(a) ⊗T_h(½(a+b))	2,34,36,39,42,46,48,52,62,64
T_v	T(a) ∧ T_v(b)	12,16,24,30,40
	T(a) ∧T_v(½(a+b))	56,58,63
T_U	T(a) ∧ T_U(b)	9,15,20,28,29
	T(a) ∧T_U(½(a+b))	54,60
T_Uh	T_U(a) ∘T_h(b)	33,43,45
T_Uv	T_U(a) ∘ T_v(b)	17
T_UU	T_U(a) ∘ T_U(b)	21
T_vv	T_v(a) ∘ T_v(b)	25,44