Affine general linear groups

The affine general linear group AGL(n,𝔽_q) is the group of invertible affine maps x ↦ Ax + b of 𝔽_qⁿ — the translations extended by the linear maps: AGL(n,𝔽_q) = 𝔽_qⁿ ⋊ GL(n,𝔽_q) (a semidirect product).

Order — |AGL(n,𝔽_q)| = qⁿ · |GL(n,𝔽_q)|.
Action — 2-transitive on the qⁿ points of affine space: the translations are transitive, and the stabilizer of 0 is GL(n,𝔽_q), transitive on the non-zero vectors.
Holomorph — for prime p, AGL(n,𝔽_p) = Hol((C_p)ⁿ), the holomorph of the elementary abelian group, since Aut((C_p)ⁿ) = GL(n,𝔽_p).

The one-dimensional case

AGL(1,𝔽_q) = 𝔽_q ⋊ 𝔽_q^× (order q(q − 1)) is sharply 2-transitive: exactly one map x ↦ ax + b carries any pair of distinct points to any other. For q > 2 it is a Frobenius group — a non-identity linear map x ↦ ax fixes only 0.

AGL(1,𝔽₃) ≅ S₃, AGL(1,𝔽₄) ≅ A₄, AGL(1,𝔽₅) = F₂₀, the Frobenius group of order 20.

Small cases beyond dimension 1

AGL(2,𝔽₂) ≅ S₄ — the affine plane over 𝔽₂ has 4 points, and every permutation of them is affine.
AGL(3,𝔽₂) — order 8 · 168 = 1344, 2-transitive on the 8 points of affine 3-space.

Over a prime-power field

For q = p^k with k > 1, AGL(n,𝔽_q) sits properly inside AGL(nk,𝔽_p). Both act on the same qⁿ = p^nk points with the same translations, but the linear part is only the 𝔽_q-linear maps GL(n,𝔽_q) within the larger 𝔽_p-linear GL(nk,𝔽_p) — so AGL(n,𝔽_q) is not a holomorph when k > 1.

AGL(1,𝔽₄) ≅ A₄ (order 12) inside AGL(2,𝔽₂) ≅ S₄ (order 24): the index 2 is exactly |GL(2,𝔽₂)| / |GL(1,𝔽₄)| = 6/3.
AGL(1,𝔽₈) — order 56, and AGL(1,𝔽₉) — order 72; both Frobenius.
AGL(2,𝔽₄) — order 2880, containing SL(2,𝔽₄) ≅ A₅, so not solvable. (Every AGL(1,𝔽_q) is solvable: 𝔽_q ⋊ 𝔽_q^× is metabelian.)

Part of the wider family of matrix groups: the point stabilizers are GL, and the affine-type 2-transitive permutation groups live between AGL(1,𝔽_q) and AGL(n,𝔽_q).

Wikipedia: Affine group