A compact G2-calibrated manifold with first Betti number b1 = 1

We construct a compact formal 7-manifold with closed $G_2$-structure and first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it such the holonomy group of associated metric is subgroup $G_2$. also associative calibrated (hence volume-minimizing) 3-tori respect to this and, for each those 3-tori, we show 3-dimensional family non-trivial deformations. fibration our over $S^2\times S^1$ generic fiber (non-calibrated) coassociative 4-torus some singular fibers.