Symmetrical 2-extensions of the 3-dimensional grid. I
For a positive integer d, a connected graph Γ is a symmetrical 2-extension of the d-dimensional grid Λd 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 Λd. The tuple (Γ, G, σ, φ) with specified components is called a realization of the symmetrical 2-extension Γ of the grid Λd. Two realizations (Γ1, G1, σ1, φ1) and (Γ2, G2, σ2, φ2) are called equivalent if there exists an isomorphism of the graph Γ1 onto Γ2 which maps σ1 onto σ2. V.I. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical 2-extensions of Λd 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 Λ2. In this work we found all, up to equivalence, realizations (Γ, G, σ, φ) of symmetrical 2-extensions of the grid Λ3 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.