Throughout this chapter, p will be a prime number. We shall show how to obtain the character tables of all groups of order pn for n ≤ 4. The method consists of examining the characters of those p-groups which contain an abelian subgroup of index p, and before explaining the method we show that all groups of order pn with 1 ≤ n ≤ 4 do, indeed, have an abelian subgroup of index p. We later give explicitly the irreducible characters of all groups of order p3 and of all groups of order 16. At the end of the chapter we point out, with references, that we have found the character tables of all groups of order less than 32.

Elementary properties of p-groups

A p-group is a group whose order is a power of the prime number p. In the first lemma we collect several well known properties of p-groups. Recall that Z(G) denotes the centre of G (see Definition 9.15).

Lemma

Let G be a group of order pn with n ≥ 1.

1. (1) If {1} ≠ H ◁ G then H ∩ Z(G) ≠ {1}. In particular, Z(G) ≠ {1}.

2. (2) If K ≤ Z(G) and G/K is cyclic, then G is abelian.

3. (3) If n ≤ 2 then G is abelian.

Proof (1) Since H ◁ G, H is a union of conjugacy classes of G, all of which have size a power of p; and H ∩ Z(G) consists of those conjugacy classes in H which have size 1.