Wreath product is defined like this
$$S_b \text{wr} S_s = (S_b)^s . S_s$$
where . means semi direct product and $S_b$ means symmetric group of size $b$. Suppose I have given the generating set of $S_b$ and $S_s$. How much time it takes to computes the wreath product?
Is there an algorithm to compute (generating set of resultant group) wreath product?
If $\langle S\rangle=G$ and $\langle T\rangle=H$ and $H \curvearrowright X$ transitively then $\langle S\cup T\rangle=G\wr_{\small X} H$, where we interpret $S$ as a subset of any one of the factors of $G$ within $G^X \rtimes H$.
This is because $S$ can be used to generate the one factor of $G$ within $G^X$, and $T$ can be used to generate all of $H$, then conjugating the factor of $G$ by elements of $H$ can get us every other copy of the group $G$ within $G^X$, and combining elements from all from the copies gives all of $G^X$.