The small Davenport constant of $H_{27}\times C_3^r$

Let $H_{27}={\mathrm{U}\mathrm{T}}_3({\mathbb{F}}_3)$ be the nonabelian group of order $27$ and exponent $3$. We prove that $\mathsf{d}(H_{27}\times C_3^r)=2r+6$ for every integer $r\ge 0$. The proof combines an affine coefficient identity in the group algebra of an elementary abelian group with a decomposition of the nonorthogonality graph of ${\mathbb{F}}_3^2$ into eight edge-disjoint zero-sum triangles. It is uniform in $r$, does not use the value of the small Davenport constant for a smaller nonabelian group, and requires no computational enumeration.