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This work applies the modular reduction technique to the Coxeter group ofrank 4 having a star diagram with labels 5, 3, and $k = 3, 4, 5, \text{ or }6$. As moduli, we use the primes in the quadratic integer ring$\mathbb{Z}[\tau]$, where $\tau = \frac{1 + \sqrt{5}}{2}$, the golden ratio. Weprove that each reduced group is a C-group, regardless of the prime used in thereduction. We also classify each reduced group as a reflection group over afinite field, whenever applicable.

Non-crystallographic tail-triangle C-groups of rank 4 and interlacing number 2