TITLE: Groups in which all characters are rational.
QUESTION [27 upvotes]: The Symmetric groups $S_n$ has interesting property that all complex irreducible characters are rational  (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in S_n$). 
Question: What are other families of (finite) groups where all complex irreducible characters are rational? Are such (finite) groups characterised?

REPLY [16 votes]: All groups that be constructed from symmetric groups via cross products and wreath products have this property.  See Section 3 of my paper "Mass formulas for local Galois representations to wreath products and cross products" http://arxiv.org/pdf/0804.4679v1.pdf
So, for example, $((S_7 \wr S_4) × S_3) \wr S_8$ has a rational character table.  In fact, taking cross products and wreathing with $S_n$ preserves the property you are asking about (see above reference).
This includes several of the examples given: $(\mathbb Z/2\mathbb Z)^n,$  hyperoctahedral groups, and Sylow 2-subgroups of $S_n$.  I am not sure if the index 2 subgroups of hyperoctahedral groups can be constructed from symmetric groups via cross products and wreath products.