  A good knowledge of the Jordan canonical form (JCF) for a tensor product of
`Jordan blocks' is key to understanding the actions of $p$-groups of matrices
in characteristic $p$. The JCF corresponds to a certain partition which depends
on the characteristic $p$, and the study of these partitions dates back to
Aitken's work in 1934. Equivalently each JCF corresponds to a certain
permutation $\pi$ introduced by Norman in 1995. These permutations $\pi =
\pi(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$.
We give necessary and sufficient conditions for $\pi(r,s,p)$ to be trivial,
building on work of M.J. Barry. We show that when $\pi(r,s,p)$ is nontrivial,
it is an involution involving reversals. Finally, we prove that the group
$G(r,p)$ generated by $\pi(r,s,p)$ for all $s$, `factors' as a wreath product
corresponding to the factorisation $r=ab$ as a product of its $p'$-part $a$ and
$p$-part $b$: precisely $G(r, p)={\sf S}_a\wr {\sf D}_b$ where ${\sf S}_a$ is a
symmetric group of degree $a$, and ${\sf D}_b$ is a dihedral group of degree
$b$.
