Determination of the stably free cancellation property for orders

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $\mathrm{\Lambda }$ be an ${\mathcal{O}}_K$-order in $A$. We give practical algorithms that determine whether $\mathrm{\Lambda }$ has stably free cancellation (SFC). As an application, we determine all finite groups $G$ of order at most $383$ such that the integral group ring $\mathbb{Z}[G]$ has SFC.