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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