  A quantum field theory with a finite abelian symmetry $G$ may be equipped
with a non-invertible duality defect associated with gauging $G$. For certain
$G$, duality defects admit an alternative construction where one starts with
invertible symmetries with certain 't Hooft anomaly, and gauging a
non-anomalous subgroup. This special type of duality defects are termed group
theoretical. In this work, we determine when duality defects are group
theoretical, among $G=\mathbb{Z}_N^{(0)}$ and $\mathbb{Z}_N^{(1)}$ in $2$d and
4d quantum field theories, respectively. A duality defect is group theoretical
if and only if its Symmetry TFT is a Dijkgraaf-Witten theory, and we argue that
this is equivalent to a certain stability condition of the topological boundary
conditions of the $G$ gauge theory. By solving the stability condition, we find
that a $\mathbb{Z}_N^{(0)}$ duality defect in 2d is group theoretical if and
only if $N$ is a perfect square, and under certain assumptions a
$\mathbb{Z}_N^{(1)}$ duality defect in 4d is group theoretical if and only if
$N=L^2 M$ where $-1$ is a quadratic residue of $M$. For these subset of $N$, we
construct explicit topological manipulations that map the non-invertible
duality defects to invertible defects. We also comment on the connection
between our results and the recent discussion of obstruction to
duality-preserving gapped phases.
