Geodesic
Symmetrical $2$-extensions of the $2$-dimensional grid. II
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 192-211 Cet article a éte moissonné depuis la source Math-Net.Ru

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The investigation of symmetrical $q$-extensions of a $d$-dimensional cubic grid $\Lambda^{d}$ is of interest both for group theory and for graph theory. For small $d\geq 1$ and $q>1$ (especially for $q=2$), symmetrical $q$-extensions of $\Lambda^{d}$ are of interest for molecular crystallography and some phisycal theories. Earlier V. Trofimov proved that there are only finitely many symmetrical $2$-extensions of $\Lambda^{d}$ for any positive integer $d$. This paper is the second and concluding part of our work devoted to the description of all, up to equivalence, realizations of symmetrical $2$-extensions of $\Lambda^{2}$ (we show that there are $162$ such realizations). In the first part of our work, which was published earlier, we found all, up to equivalence, realizations of symmetrical $2$-extensions of $\Lambda^{2}$ such that only the trivial automorphism fixes all blocks of the imprimitivity system ($87$ realizations). In the present paper, we find the remaining realizations of symmetrical $2$-extensions of $\Lambda^{2}$.
Keywords: symmetrical extension of a graph, $d$-dimensional grid. 
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E. A. Konoval'chik; K. V. Kostousov. Symmetrical $2$-extensions of the $2$-dimensional grid. II. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 192-211. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a18/

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