Strong generating set

Let $G\leq S_{n}$ be a permutation group. Let $B=(\beta _{1},\beta _{2},\ldots ,\beta _{r})$ be a sequence of distinct integers, $\beta _{i}\in \{1,2,\ldots ,n\}$, such that the pointwise stabilizer of $B$ is trivial. Define $B_{i}=(\beta _{1},\beta _{2},\ldots ,\beta _{i})$, and define $G^{(i)}$ to be the pointwise stabilizer of $B_{i}$. A strong generating set for the base $B$ is a set $S\subset G$ such that $\langle S\cap G^{(i)}\rangle =G^{(i)}$ for each $1\leq i\leq r$.

The base and the SGS are said to be non-redunant if $G^{(i)}\neq G^{(j)}$ for $i\neq j$.