  Permutation codes of length $n$ and distance $d$ is a set of permutations on
$n$ symbols, where the distance between any two elements in the set is at least
$d$. Subgroup permutation codes are permutation codes with the property that
the elements are closed under the operation of composition. In this paper,
under the distance metric $\ell_{\infty}$-norm, we prove that finding the
minimum weight codeword for subgroup permutation code is NP-complete. Moreover,
we show that it is NP-hard to approximate the minimum weight within the factor
$7/6-\epsilon$ for any $\epsilon>0$.
