### Abstract

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H_{4}) of order 14,400. Its derived subgroup is the largest finite subgroup W(H_{4})/Z_{2} of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H_{2}) × W(H_{2})]⋊ Z_{4} and W(H_{3}) × Z_{2} possess non-crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H_{4}) with sets of quaternion pairs acting on the quaternionic root systems.

Original language | English (US) |
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Pages (from-to) | 441-452 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 412 |

Issue number | 2-3 |

DOIs | |

State | Published - Jan 15 2006 |

Externally published | Yes |

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### Keywords

- Coxeter groups
- Quaternions
- Structure of groups
- Subgroup structure

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

_{4}) and quaternions.

*Linear Algebra and Its Applications*,

*412*(2-3), 441-452. https://doi.org/10.1016/j.laa.2005.07.018