My goal is to show that $Alt(5) \wr Sym(3)$ is a primitive permutation group of type SD.
Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH = G$). $G$ is of type SD if
- $N$ is non-abelian and the unique minimal normal subgroup (so $N \cong T^k$, where $T$ is a non-abelian simple group)
- $N \cap H$ is the diagonal subgroup $\{(t, t, ..., t): t \in T\}$
It follows (as claimed here) that $N$ acts on the cosets of $N \cap H$ in $N$, where $(t_1, ..., t_k)$ takes the coset of $(s_1, ..., s_k)$ to the coset of $(t_k^{-1}s_1t_1, ..., 1)$, so we identify $\Omega$ (the underlying set) with $T^{k-1}$. And we identify $G$ with a subgroup of $Aut(T^k) \cong Aut(T) \wr Sym(k)$.
In the case $N = Alt(5)^3$, $G =Alt(5) \wr Sym(3)$ is in the form I want. The only thing I am struggling to understand is the action of this group.
Is the action of $NH$ on the cosets of $H$ isomorphic to the action of $N$ on the cosets of $N \cap H$?
If the underlying set $\Omega$ is isomorphic to $Alt(5)^2$, how can I explicitly describe the action of $Alt(5) \wr Sym(3)$?
I am not sure how to answer the $2$nd question without assuming the first. For the first question, (since we do not assume $H$ is normal), I defined a natural bijection between $NH/H$ and $N / N \cap H$, but I could not show that this bijection respects the action.