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PGD Subgroups

We list here the 14 kaleidoscopic (or Coxeter) subgroups of *246 that are compatible with the genus-3 translational unit (ooo) for the P, G, and D surfaces. A kaleidoscopic subgroup is one with an orbifold that has a single boundary component and no other symmetry features. The P, G, and D space groups are the three dimensional symmetries of the E-tiling obtained by wrapping a U-tiling onto the respective surface.

There are a total of 131 subgroups of *246 that retain the ooo translational symmetry. A complete list is available here.

Domain	Subgroup	Normal	Index	Curvature	P space group	G space group	D space group	U-tiling Count
	*246	Y	1	-1/24	Im-3m	Ia-3d	Pn-3m	204
	*266	Y	2	-1/12	Pn-3m	Ia-3	Fd-3m	92
	*344	Y	2	-1/12	Pm-3n	I-43d	P-43m	108
	*2223	Y	2	-1/12	Pm-3m	I4(1)32	P4(2)32	336
	*2224	N	3	-1/8	I4/mmm	I4(1)/acd	P4(2)/nnm	694
	*2323	Y	4	-1/6	P4(2)32	I2(1)3	F-43m	183
	*2244a	N	6	-1/4	P4(2)/mmc	I-42d	P-42m	366
	*2244b	N	6	-1/4	P4/nmm	I4(1)/a	I4(1)/amd	370
	*22222a	N	6	-1/4	P4/mmm	I4(1)22	P4(2)22	889
	*22222b	N	6	-1/4	Fmmm	Fddd	Cmma	893
	*2626	N	8	-1/3	R-3m	R-3	R-3m	396
	*222222a	N	12	-1/2	Cmma	C2/c	Imma	1026
	*4444	N	12	-1/2	P-4m2	I-4	I-4m2	211
	*222222b	Y	12	-1/2	Pmmm	I2(1)2(1)2(1)	P222	327