Four Dimensions
The four-dimensional point groups, limiting to purely reflectional groups, can be listed by their Coxeter group, and like the polyhedral groups of 3D, can be named by their related convex regular 4-polytopes. 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 three 3-fold gyration points and symmetry order 60. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeter's notation, for example ] with its order doubled to 240.
| Coxeter group/notation | Coxeter diagram | Order | Related polytopes | ||
|---|---|---|---|---|---|
| A4 | 120 | 5-cell | |||
| A4×2 | ] | 240 | 5-cell dual compound | ||
| BC4 | 384 | 16-cell/Tesseract | |||
| D4 | 192 | Demitesseractic | |||
| D4×2 = BC4 | <> = | = | 384 | ||
| D4×6 = F4 | ] = | = | 1152 | ||
| F4 | 1152 | 24-cell | |||
| F4×2 | ] | 2304 | 24-cell dual compound | ||
| H4 | 14400 | 120-cell/600-cell | |||
| A3×A1 | 48 | Tetrahedral prism | |||
| A3×A1×2 | ,2] = | = | 96 | Octahedral prism | |
| BC3×A1 | 96 | ||||
| H3×A1 | 240 | Icosahedral prism | |||
| A2×A2 | 36 | Duoprism | |||
| A2×BC2 | 48 | ||||
| A2×H2 | 60 | ||||
| A2×G2 | 72 | ||||
| BC2×BC2 | 64 | ||||
| BC22×2 | ] | 128 | |||
| BC2×H2 | 80 | ||||
| BC2×G2 | 96 | ||||
| H2×H2 | 100 | ||||
| H2×G2 | 120 | ||||
| G2×G2 | 144 | ||||
| I2(p)×I2(q) | 4pq | ||||
| I2(2p)×I2(q) | ,2,q] = | = | 8pq | ||
| I2(2p)×I2(2q) | ],2,] = | = | 16pq | ||
| I2(p)2×2 | ] | 8p2 | |||
| I2(2p)2×2 | ,2,]] = ] | = | 32p2 | ||
| A2×A1×A1 | 24 | ||||
| BC2×A1×A1 | 32 | ||||
| H2×A1×A1 | 40 | ||||
| G2×A1×A1 | 48 | ||||
| I2(p)×A1×A1 | 8p | ||||
| I2(2p)×A1×A1×2 | ,2,2] = | = | 16p | ||
| I2(p)×A12×2 | ] = | = | 16p | ||
| I2(2p)×A12×4 | ],2,] = | = | 32p | ||
| A1×A1×A1×A1 | 16 | 4-orthotope | |||
| A12×A1×A1×2 | ,2,2] = | = | 32 | ||
| A12×A12×4 | ],2,] = | = | 64 | ||
| A13×A1×6 | ,2] = | = | 96 | ||
| A14×24 | ] = | = | 384 | ||
Read more about this topic: Point Group
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