  Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed
with $[n]=\{1,2,\dots,n\}$ and the rows are indexed with $[n]^3$. The row
corresponding to $(x_1,x_2,x_3)\in [n]^3$ is given by $\sum_{i=1}^3 e_{x_i}$,
where $e_1,e_2,\dots,e_n$ is the standard basis of $\mathbb{R}^{[n]}$. Let
$A_n$ be random $n\times n$ submatrix of $B_n$, where the probability that we
choose a submatrix $C$ is proportional to $|\det(C)|^2$.
  Let $p\ge 5$ be a prime. We prove that the asymptotic distribution of the
$p$-Sylow subgroup of the cokernel of $A_n$ is given by the Cohen-Lenstra
heuristics.
  Our result is motivated by the conjecture that the first homology group of a
random two dimensional hypertree is also Cohen-Lenstra distributed.
