Special semilinear groups

The special semilinear group ΣL(n,𝔽_q) is the special linear group extended by the field automorphisms:

ΣL(n,𝔽_q) = SL(n,𝔽_q) ⋊ Gal(𝔽_q / 𝔽_p) ≅ SL(n,𝔽_q) ⋊ C_d, where q = p^d — the maps x ↦ A·x^σ of 𝔽_qⁿ with det A = 1.

Order — |ΣL(n,𝔽_q)| = d · |SL(n,𝔽_q)|.
Inside ΓL — ΣL(n,𝔽_q) is normal in ΓL(n,𝔽_q) with ΓL / ΣL ≅ GL / SL ≅ C_q−1. (The determinant is not a homomorphism on ΓL: a field twist sends det A to (det A)^σ. The quotient is still cyclic of order q − 1, since ΓL = GL · ΣL and GL ∩ ΣL = SL.)
Over a prime field — ΣL(n,𝔽_p) = SL(n,𝔽_p), the Galois group being trivial.
Dimension 1 — SL(1,𝔽_q) = 1, so ΣL(1,𝔽_q) ≅ C_d: the Galois group alone.

Small cases

ΣL(2,𝔽₄) ≅ S₅ (order 120): SL(2,𝔽₄) ≅ A₅, and the Frobenius acts as the outer automorphism.
ΣL(2,𝔽₈) — order 3 · 504 = 1512.
ΣL(2,𝔽₉) — order 2 · 720 = 1440.
ΣL(3,𝔽₄) — order 2 · 60 480 = 120 960.

One group, several names

Over an even field gcd(2, q − 1) = 1, so SL(2,𝔽_q) has trivial center and SL(2,𝔽_q) = PSL(2,𝔽_q) = PGL(2,𝔽_q). Adjoining the same Galois group to all three collapses the names too:

ΣL(2,𝔽_q) = PΣL(2,𝔽_q) = PΓL(2,𝔽_q) for q even.

So ΣL(2,𝔽₄), PΣL(2,𝔽₄) and PΓL(2,𝔽₄) are three names for S₅ — the catalog lists such a group under each name it answers to.

Part of the wider family of matrix groups: the determinant-1 part of ΓL, with projective quotient PΣL and affine extension AΣL.

Wikipedia: Semilinear map