# Symmetrical 2-extensions of the 3-dimensional grid. I

### Abstract

For a positive integer *d*, a connected graph *Γ* is a symmetrical 2-extension of the *d*-dimensional grid *Λ*^{d} if there exists a vertex-transitive group *G* of automorphisms of *Γ* and its imprimitivity system *σ* with blocks of size 2 such that there exists an isomorphism *φ* of the quotient graph *Γ*/*σ* onto *Λ*^{d}. The tuple (*Γ*, *G*, *σ*, *φ*) with specified components is called a realization of the symmetrical 2-extension *Γ* of the grid *Λ*^{d}. Two realizations (*Γ*_{1}, *G*_{1}, *σ*_{1}, *φ*_{1}) and (*Γ*_{2}, *G*_{2}, *σ*_{2}, *φ*_{2}) are called equivalent if there exists an isomorphism of the graph *Γ*_{1} onto *Γ*_{2} which maps *σ*_{1} onto *σ*_{2}. V.I. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical 2-extensions of *Λ*^{d} for each positive integer *d*. E.A. Konovalchik and K.V. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid *Λ*^{2}. In this work we found all, up to equivalence, realizations (*Γ*, *G*, *σ*, *φ*) of symmetrical 2-extensions of the grid *Λ*^{3} for which only the trivial automorphism of *Γ* preserves all blocks of *σ*. Namely we prove that there are 5573 such realizations, and that among corresponding graphs *Γ* there are 5350 pairwise non-isomorphic.