### 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 |
---|---|

Pages (from-to) | 441-452 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 412 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Jan 15 2006 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

*Linear Algebra and Its Applications*,

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

**Maximal subgroups of the Coxeter group W(H4) and quaternions.** / Koca, Mehmet; Koç, Ramazan; Al-Barwani, Muataz; Al-Farsi, Shadia.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 412, no. 2-3, pp. 441-452. https://doi.org/10.1016/j.laa.2005.07.018

}

TY - JOUR

T1 - Maximal subgroups of the Coxeter group W(H4) and quaternions

AU - Koca, Mehmet

AU - Koç, Ramazan

AU - Al-Barwani, Muataz

AU - Al-Farsi, Shadia

PY - 2006/1/15

Y1 - 2006/1/15

N2 - The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 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(H2) × W(H2)]⋊ Z4 and W(H3) × Z2 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(H4) with sets of quaternion pairs acting on the quaternionic root systems.

AB - The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 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(H2) × W(H2)]⋊ Z4 and W(H3) × Z2 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(H4) with sets of quaternion pairs acting on the quaternionic root systems.

KW - Coxeter groups

KW - Quaternions

KW - Structure of groups

KW - Subgroup structure

UR - http://www.scopus.com/inward/record.url?scp=27844499287&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27844499287&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2005.07.018

DO - 10.1016/j.laa.2005.07.018

M3 - Article

AN - SCOPUS:27844499287

VL - 412

SP - 441

EP - 452

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 2-3

ER -