Relations for a class of terminating ${}_4F_3(4)$ hypergeometric series

We derive relations for a certain class of terminating ${}_4F_3(4)$ hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group $S_3$. We further study relations for terminating ${}_3F_2(4)$ series that fall under two families. By using a series reversal, we examine the corresponding terminating ${}_4F_3(1/4)$ and ${}_3F_2(1/4)$ series relations. We additionally derive formulas for the sums of the first $n+1$ terms of several nonterminating ${}_3F_2(4)$ and ${}_3F_2(1/4)$ series. We also show how certain known summation formulas for terminating ${}_2F_1(4)$ and ${}_3F_2(4)$ series follow as limiting cases of some of our relations.