Projective symplectic groups

The projective symplectic group PSp(2m,𝔽_q) is the symplectic group Sp(2m,𝔽_q) modulo its center {±I}.

Order — |PSp(2m,𝔽_q)| = |Sp(2m,𝔽_q)| / gcd(2, q − 1) — the center has order 2 for odd q and is trivial for even q (where PSp = Sp).
Simple — for every m ≥ 1 and prime power q except PSp(2,𝔽₂) ≅ S₃, PSp(2,𝔽₃) ≅ A₄, and PSp(4,𝔽₂) ≅ S₆ — the classical simple groups of Lie type Cₘ.
Dimension 2 — PSp(2,𝔽_q) = PSL(2,𝔽_q), inheriting the PSL exceptions.

PSp(2,𝔽₅) = PSL(2,𝔽₅) ≅ A₅ (order 60).
PSp(4,𝔽₂) = Sp(4,𝔽₂) ≅ S₆ (order 720) — not simple; its derived subgroup A₆ ≅ PSL(2,𝔽₉) is.
PSp(4,𝔽₃) — order 25 920, simple; exceptionally isomorphic to PSU(4,𝔽₂), the unitary group of the same order.

Part of the wider family of matrix groups.