Unitriangular groups

The unitriangular group UT(n,𝔽_q) is the group of upper-triangular n × n matrices over 𝔽_q with 1s on the diagonal.

Order — |UT(n,𝔽_q)| = q^n(n−1)/2, one field entry per position above the diagonal.
Sylow subgroup — for q = pᵈ, UT(n,𝔽_q) is a Sylow p-subgroup of GL(n,𝔽_q): it carries the full p-part of the order formula.
Nilpotent of class n − 1 — the lower central series γ_k consists of the matrices supported from the k-th superdiagonal outward, and the upper central series climbs the same ladder in reverse. The center is the corner: matrices differing from I only in the (1, n) entry, isomorphic to (𝔽_q, +).
Universal for p-groups — every finite p-group embeds in UT(n,𝔽_p) for some n (embed in GL(m,𝔽_p) by the regular representation, then move into a Sylow p-subgroup).

UT(2,𝔽_q) ≅ (𝔽_q, +), elementary abelian.
UT(3,𝔽_q) is the Heisenberg group H(3,𝔽_q) — the two families name the same group at n = 3, and only there. For q = 2 it is the dihedral group D₄ of order 8; for an odd prime q, the extraspecial group of exponent q.
UT(4,𝔽₂) — order 64, class 3.

Part of the wider family of matrix groups: the unipotent part of the Borel subgroup of GL.