Does there exist some sort of classification of incompressible groups?

Question

It is well known, that any finite group of order $n$ is isomorphic to a subgroup of $S_n$. Let’s call a finite group $G$ incompressible iff it is not isomorphic to any subgroup of $S_{|G|-1}$ . Does there exist some sort of classification of incompressible groups?

What I currently know:

Any non-trivial incompressible group has non-trivial center

If the center of a group $G$ is trivial, then it acts faithfully by conjugation on $G \setminus \{e\}$.

If an incompressible group is non-trivially decomposed into a direct product of two its subgroups, it is isomorphic to $C_2 \times C_2$

One can construct a faithful action of $H \times K$ on $H \cup K$. It is defined as $(h, k)h_0 \mapsto hh_0$ and $(h, k)h_0 \mapsto kk_0$ for $h, h_0 \in H$, $k, k_0 \in K$.

$|H| + |K| \geq |H||K|$ iff either one of the groups is trivial, or both of them are isomorphic to $C_2$.

$C_2 \times C_2$ is the only possible group and indeed is not contained in $S_3$.

I also conjecture, that «direct product» in this statement can be replaced with «semidirect product», but do not know how to prove that.

All cyclic $p$-groups are incompressible

If $p$ is prime, then $S_{p^n - 1}$ does not have an element of order $p^n$

$Q_8$ is incompressible

$S_7$ does not contain $Q_8$ as a subgroup

Answer 1

These were fully classified by Johnson in the paper 'Minimal Permutation Representations of Finite Groups'.

A group is incompressible iff it is isomorphic to one of the following:

• Cyclic group of prime power order $C_{p^n}$
• Generalised quaternion $2$-group $\langle x,y|x^{2^n}=1,x^{2^{n-1}}=y^2,x^y=x^{-1}\rangle$
• the Klein four-group $C_2\times C_2$

The proof is reasonably short so well worth looking up!

Reference: Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. MR 316540 DOI: 10.2307/2373739.