# E.A. Konoval’chik, K.V. Kostousov. Symmetrical 2-extensions of the 2-dimensional grid. II ... P. 192-211

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.

The paper was received by the Editorial Office on November 12, 2016

Elena Aleksandrovna Konoval’chik, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics
and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia;
Nosov Magnitogorsk State Technical University, Magnitogorsk, 455000 Russia,
e-mail: asmi@imm.uran.ru .

Kirill Viktorovich Kostousov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and
Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: asmi@imm.uran.ru .

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