In general, the construction of subspace codes or, in particular, cyclic Grassmannian codes in some projective space P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n) is highly mathematical and requires substantial computational power for the resulting searches. In this paper, we present a new method for the construction of cyclic Grassmannian codes. To do that was designed and implemented a series of algorithms using the GAP System for Computational Discrete Algebra and Wolfram Mathematica software. We also present a classification of such codes in the space P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n), with n at most 9. The fundamental idea to construct and classify the cyclic Grassmannian codes is to endow the projective space P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n) with a graph structure and then find cliques.