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Maximal Subgroups of the Coxeter Group $W(H_4)$ and Quaternions

Mehmet Koca·Ramazan Koc·Muataz Al-Barwani·Shadia Al-Farsi
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Abstract

The largest finite subgroup of O(4) is the noncrystallographic Coxeter group $W(H_{4})$ of order 14400. 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 $[ W(H_{2})\times W(H_{2})] \times Z_{4}$ and $W(H_{3})\times Z_{2}$ possess noncrystallographic 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 $SU(3)\times SU(3)$%, 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.

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Mehmet Koca, Ramazan Koc, Muataz Al-Barwani, Shadia Al-Farsi. 2005-10-22. Maximal Subgroups of the Coxeter Group $W(H_4)$ and Quaternions. https://arxiv.org/abs/hep-th/0510191

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