Five Dimensions
The five-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 four 3-fold gyration points and symmetry order 360.
| Coxeter group | Coxeter diagrams |
Order | Related regular/prismatic polytopes | ||
|---|---|---|---|---|---|
| A5 | 720 | 5-simplex | |||
| A5×2 | ] | 1440 | 5-simplex dual compound | ||
| BC5 | 3840 | 5-cube, 5-orthoplex | |||
| D5 | 1920 | 5-demicube | |||
| D5 | = | 3840 | |||
| A4×A1 | 240 | 5-cell prism | |||
| A4×A1×2 | ,2] | 480 | |||
| BC4×A1 | 768 | tesseract prism | |||
| F4×A1 | 2304 | 24-cell prism | |||
| F4×A1×2 | ,2] | 4608 | |||
| H4×A1 | 28800 | 600-cell or 120-cell prism | |||
| D4×A1 | 384 | Demitesseract prism | |||
| A3×A2 | 144 | Duoprism | |||
| A3×A2×2 | ,2,3] | 288 | |||
| A3×BC2 | 192 | ||||
| A3×H2 | 240 | ||||
| A3×G2 | 288 | ||||
| A3×I2(p) | 48p | ||||
| BC3×A2 | 288 | ||||
| BC3×BC2 | 384 | ||||
| BC3×H2 | 480 | ||||
| BC3×G2 | 576 | ||||
| BC3×I2(p) | 96p | ||||
| H3×A2 | 720 | ||||
| H3×BC2 | 960 | ||||
| H3×H2 | 1200 | ||||
| H3×G2 | 1440 | ||||
| H3×I2(p) | 240p | ||||
| A3×A12 | 96 | ||||
| BC3×A12 | 192 | ||||
| H3×A12 | 480 | ||||
| A22×A1 | 72 | duoprism prism | |||
| A2×BC2×A1 | 96 | ||||
| A2×H2×A1 | 120 | ||||
| A2×G2×A1 | 144 | ||||
| BC22×A1 | 128 | ||||
| BC2×H2×A1 | 160 | ||||
| BC2×G2×A1 | 192 | ||||
| H22×A1 | 200 | ||||
| H2×G2×A1 | 240 | ||||
| G22×A1 | 288 | ||||
| I2(p)×I2(q)×A1 | 8pq | ||||
| A2×A13 | 48 | ||||
| BC2×A13 | 64 | ||||
| H2×A13 | 80 | ||||
| G2×A13 | 96 | ||||
| I2(p)×A13 | 16p | ||||
| A15 | 32 | 5-orthotope | |||
Read more about this topic: Point Group
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