Iterated wreath products appear naturally as Sylow subgroups of symmetric groups of prime power degree.
Note that iterated wreath products grow quite rapidly.
The wreath product construction is not commutative; notice that even the order is different.
Note that the regular wreath product is also different in this case, since the second argument here is not the regular permutation representation of the symmetric group.
Since both and are transitive, so too is their wreath product.
However, in general, the wreath product is not primitive.
Here we construct a wreath product with an intransitive second argument.
The resulting group is not transitive.
Here we construct a wreath product with an intransitive first argument.
Again, the result is an intransitive group.
Notice that, if a group is already regular, then its regular wreath product is isomorphic to the ordinary wreath product.
