We show that the modular group γ3on the 27-dimensional exceptional domain admits only one multiplier system. At first we demonstrate that ¡aequals its own commutator subgroup by Jordan theoretic means, where the main tool is Hua's identity. Maaβ' description of all multiplier systems for the Siegel modular group then leads to the result. As a by-product we can describe the multiplier systems for the modular groups on the 10-dimensional boundary component and the half-spaces of quaternions.