Six Dimensions
The six-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group. Related pure rotational groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example + has five 3-fold gyration points and symmetry order 2520.
| Coxeter group | Coxeter diagram |
Order | Related regular/prismatic polytopes | |
|---|---|---|---|---|
| A6 | 5040 (7!) | 6-simplex | ||
| A6×2 | ] | 10080 (2×7!) | 6-simplex dual compound | |
| BC6 | 46080 (26×6!) | 6-cube, 6-orthoplex | ||
| D6 | 23040 (25×6!) | 6-demicube | ||
| E6 | 51840 (72×6!) | 122, 221 | ||
| A5×A1 | 1440 (2×6!) | 5-simplex prism | ||
| BC5×A1 | 7680 (26×5!) | 5-cube prism | ||
| D5×A1 | 3840 (25×5!) | 5-demicube prism | ||
| A4×I2(p) | 240p | Duoprism | ||
| BC4×I2(p) | 768p | |||
| F4×I2(p) | 2304p | |||
| H4×I2(p) | 28800p | |||
| D4×I2(p) | 384p | |||
| A4×A12 | 480 | |||
| BC4×A12 | 1536 | |||
| F4×A12 | 4608 | |||
| H4×A12 | 57600 | |||
| D4×A12 | 768 | |||
| A32 | 576 | |||
| A3×BC3 | 1152 | |||
| A3×H3 | 2880 | |||
| BC32 | 2304 | |||
| BC3×H3 | 5760 | |||
| H32 | 14400 | |||
| A3×I2(p)×A1 | 96p | Duoprism prism | ||
| BC3×I2(p)×A1 | 192p | |||
| H3×I2(p)×A1 | 480p | |||
| A3×A13 | 192 | |||
| BC3×A13 | 384 | |||
| H3×A13 | 960 | |||
| I2(p)×I2(q)×I2(r) | 8pqr | Triaprism | ||
| I2(p)×I2(q)×A12 | 16pq | |||
| I2(p)×A14 | 32p | |||
| A16 | 64 | 6-orthotope | ||
Read more about this topic: Point Group
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—Henry David Thoreau (1817–1862)
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