W(1) as W(n + 1). Then W(n) is the Sylow p-subgroup of the symmetric group Sym(pn). Maximal p-subgroups of the general linear group GL(n,Q) are direct products of various W(n). It has order pk where k = (pn − 1)/(p − 1). It has nilpotency class pn−1, and its lower central series, upper central series, lower exponent-p central series, and upper exponent-p central series are equal. It is generated by its elements of order p, but its exponent is pn. The second such group, W(2), is also a p-group of maximal class, since it has order pp+1 and nilpotency class p, but is not a regular p-group. Since groups of order pp are always regular groups, it is also a minimal such example. Generalized dihedral groups When p = 2 and n = 2, W(n) is the dihedral group of order 8, so in some sense W(n) provides an analogue for the dihedral group for all primes p when n = 2. However, for higher n the analogy becomes strained. There is a different family of examples that more closely mimics the dihedral groups of order 2n, but that requires a bit more setup. Let ζ denote a primitive pth root of unity in the complex numbers, let Z[ζ] be the ring of cyclotomic integers generated by it, and let P be the prime ideal generated by 1−ζ. Let G be a cyclic group of order p generated by an element z. Form the semidirect product E(p) of Z[ζ] and G where z acts as multiplication by ζ. The powers Pn are normal subgroups of E(p), and the example groups are E(p,n) = E(p)/Pn. E(p,n) has order pn+1 and nilpotency class n, so is a p-group of maximal class. When p = 2, E(2,n) is the dihedral group of order 2n. When p is odd, both W(2) and E(p,p) are irregular groups of maximal class and order pp+1, but are not isomorphic. Unitriangular matrix groups The Sylow subgroups of general linear groups are another fundamental family of examples. Let V be a vector space of dimension n with basis { e1, e2, ..., en } and define Vi to be the vector space generated by { ei, ei+1, ..., en } for 1 ≤ i ≤ n, and define Vi = 0 when i > n. For each 1 ≤ m ≤ n, the set of invertible linear transformations of