Let \(k\) be an algebraically closed field of characteristic 3. \(\text{GL}(4,k)\) has a 16-dimensional irreducible module \(V\) of highest weight \(\lambda_1+\lambda_2\) which can be obtained as \(V=S^3(k^4)/\{x^3\mid x\in S^1(k^4)\}\). This module is one of the open cases in a classification of irreducible modules for almost simple algebraic groups over an algebraically closed field of positive characteristic for which there are a finite number of orbits on points. By using computer algebra the authors prove that there are exactly 10 \(\text{GL}(4)\)-orbits of vectors in \(V\setminus\{0\}\) and list representative vectors for these orbits.