Example of a primitive group of affine type and of twisted wreath product type

Question

Good evening, According to the O'Nan-Scott theorem, primitive finite group are classified into five classes: affine type, product type, almost simple type, diagonal type and twisted wreath product type. Maybe it is trivial, but I am looking for an example of a primitive group of affine type and another one of twisted wreath product type. I could find them nowhere. Thanks for reading.

Answer 1

As I said in my comment, there are lots of small examples of primitive groups of affine type, like $S_3$ (or even $S_2$). They all arise as subgroups of the affine groups ${\rm AGL}(n,p)$, which is a semidirect product $p^n\!:\!{\rm GL}(n,p)$, and the normal elementary abelian subgroup $N$ of order $p^n$ is a regular normal subgroup in the primitive action. For the group to be primitive, the subgroup of ${\rm GL}(n,p)$ must act irreducibly on $N$. Another simple class of examples are cyclic groups of prime order $p$ in their regular representations, and the dihedral groups of order $2p$.

The primitive groups of twisted wreath product type are much harder to find! The smallest such group is a permutation representation of $A_5 \wr A_6$ on the cosets of a subgroup $H$ isomorphic to $A_6$, and has degree $60^6$. (The base group of the wreath product is a regular normal subgroup in the primitive action.)

The subgroup $H$ is a complement to the base group in the wreath product, but it is not a conjugate of the natural complement.

The natural complement simply permutes the six copies of $A_5$ in the base group. In fact $H$ also permutes therm but the point stabilizer $A_5$ in $H$ induces the group of inner automorphisms of the copy of $A_5$ in the base group that it stabilizes. Hence the name twisted wreath product. (When I first heard of twisted wreath products, I assumed that they were some variant of wreath products, but in fact as abstract groups they are isomorphic to the normal wreath products. It is the action of the complement on the base group that is twisted.)

I recommend Dixon and Mortimer's book on Permutation Groups for a detailed treatment. You can also find it in Peter Cameron's book on Permutation groups, but that is more concise.