  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.
