The Parker loop has order $2^{13}$, and reducing it modulo its 'center' yields -- not just a $12$-dimensional $\mathbb{Z} /(2)$-vector space, but one which can be naturally identified with the subspace of $(\mathbb{Z} /(2))^{24}$ which is the extended $24$-bit Golay code.

Analogous to the extended $24$-bit Golay code is the $12$-coordinate extended ternary Golay code, which is a $6$-dimensional subspace of $(\mathbb{Z} /(3))^{12}$. Is there likewise a known analogue of the Parker loop which has order $3^{7}$ and which may be likewise identified with the extended ternary Golay code once it quotiented by its center? (And, if no such thing is known, is it because it has been proven not to exist?)