  We study a natural random walk on the $n \times n$ upper triangular matrices,
with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or
subtract a uniformly random row to the row above. We show that the mixing time
of this random walk is $O(m^2n \log n+ n^2 m^{o(1)})$. This answers a question
of Stong and of Arias-Castro, Diaconis, and Stanley.
