Symmetry

Truncated order-4 hexagonal tiling with *662 mirror lines

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1⁺,6,1⁺,6,1⁺] (3333) is the commutator subgroup of [6,6].

Larger subgroup constructed as [6,6^*], removing the gyration points of (6*3), index 12 becomes (*333333).

The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.

Small index subgroups of [6,6] (*662)
Index	1	2			4
Diagram
Coxeter	[6,6]	[1+,6,6] =	[6,6,1+] =	[6,1+,6] =	[1+,6,6,1+] =	[6+,6+]
Orbifold	*662	*663		*3232	*3333	33×
Direct subgroups
Diagram
Coxeter		[6,6+]	[6+,6]	[(6,6,2+)]	[6,1+,6,1+] = = = =	[1+,6,1+,6] = = = =
Orbifold		6*3		2*33	3*33
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[6,6]+	[6,6+]+ =	[6+,6]+ =	[6,1+,6]+ =	[6+,6+]+ = [1+,6,1+,6]+ = = =
Orbifold	662	663		3232	3333
Radical subgroups
Index		12		24
Diagram
Coxeter		[6,6*]	[6*,6]	[6,6*]+	[6*,6]+
Orbifold		*333333		333333