We investigate a certain linear combination $K(vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V={(a,b,c,d,e,f,g)inBbb C^7colon e+f+g-a-b-c-d=1}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $mu_1,mu_2,mu_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(mu_1vec{x})$, $K(mu_2vec{x})$, and $K(mu_3vec{x})$, provided no two of the $mu_j$s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|atop 3})=({32atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(mu_1,mu_2,mu_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $mu_j$s, which in turn is defined in terms of the expression of the $mu_j$s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.
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