Affine general semilinear groups

The affine general semilinear group AΓL(n,𝔽_q) is the group of maps x ↦ A·x^σ + b of 𝔽_qⁿ — affine maps allowed a field twist. It is the translations extended by ΓL(n,𝔽_q):

AΓL(n,𝔽_q) = 𝔽_qⁿ ⋊ ΓL(n,𝔽_q) (a semidirect product), q = p^d.

Order — |AΓL(n,𝔽_q)| = qⁿ · d · |GL(n,𝔽_q)| = d · |AGL(n,𝔽_q)|.
Over a prime field — AΓL(n,𝔽_p) = AGL(n,𝔽_p).
Action — 2-transitive on the qⁿ points of affine space: the translations are transitive, and the stabilizer of 0 is ΓL(n,𝔽_q), already transitive on the non-zero vectors through its GL(n,𝔽_q) part.
Inside a bigger affine group — semilinear maps are 𝔽_p-affine, so AΓL(n,𝔽_q) ≤ AGL(nd,𝔽_p) = Hol((C_p)^nd), the holomorph of the elementary abelian group of the same order.

The one-dimensional case

AΓL(1,𝔽_q) = {x ↦ a·x^σ + b}, order q(q − 1)d — the largest group of "affine" maps of a finite field.

AΓL(1,𝔽₄) ≅ S₄ (order 24) — all permutations of the four points of 𝔽₄.
AΓL(1,𝔽₈) — order 8 · 7 · 3 = 168, solvable and 2-transitive on 8 points (a different group of order 168 from the simple PSL(2,𝔽₇)).
AΓL(1,𝔽₉) — order 9 · 8 · 2 = 144.
AΓL(1,𝔽₁₆) — order 16 · 15 · 4 = 960.

Which subgroups stay 2-transitive? A subgroup of AΓL(1,𝔽_q) containing all translations is 𝔽_q ⋊ H with H ≤ ΓL(1,𝔽_q), and it is 2-transitive exactly when H is transitive on 𝔽_q^×. These are the 1-dimensional affine 2-transitive groups; the smallest is AGL(1,𝔽_q) = 𝔽_q ⋊ 𝔽_q^×, the sharply 2-transitive Frobenius one.

Higher dimensions

AΓL(2,𝔽₄) — order 16 · 2 · 180 = 5760.
AΓL(2,𝔽₉) — order 81 · 2 · 5760 = 933 120.

Part of the wider family of matrix groups: AGL with a field twist, and the translation-free part is ΓL.

Wikipedia: Affine group