  We study the group of transformations of 4F3 hypergeometric functions
evaluated at unity with one unit shift in parameters. We reveal the general
form of this family of transformations and its group property. Next, we use
explicitly known transformations to generate a subgroup whose structure is then
thoroughly studied. Using some known results for 3F2 transformation groups, we
show that this subgroup is isomorphic to the direct product of the symmetric
group of degree 5 and 5-dimensional integer lattice. We investigate the
relation between two-term 4F3 transformations from our group and three-term 3F2
transformations and present a method for computing the coefficients of the
contiguous relations for 3F2 functions evaluated at unity. We further furnish a
class of summation formulas associated with the elements of our group. In the
appendix to this paper, we give a collection of Wolfram Mathematica routines
facilitating the group calculations.
