Heitler–Heisenberg multispin states were studied via irreducible representations of the united symmetry with respect to permutations and space transformations group. The mean energy is given in explicit form in terms of the characters of the joint group irreducible representations. The system’s Fockian covariance incorporates its exchange integral of the self-consistent states into the Heisenberg chain theory. External fields account is delivered in perturbation theory frame. Its application to statistical physics approach leads to the thermodynamic parameter evaluation. The nanotube example with space symmetry including rotations and translations, is studied. Its symmetry introduces basic closest neighbor exchange integrals that enter the statistical sum.