TITLE: Which groups have strictly rational representations?
QUESTION [16 upvotes]: It can be shown, via the construction of the representations of the symmetric group, that every representation of $S_n$ is equivalent to a representation with values in $\mathbb{Q}.$ Presumably, this is a fairly rare phenomenon: it clearly doesn't hold for cyclic groups ($\mathbb{Z}/p\mathbb{Z}$ has one-dimensional representations given by the $p$th roots of unity, hence $p-1$ of its representations lie outside of $\mathbb{Q}$).
Moreover, there is a formula which constructs the rational characters of a group (due to Artin: see Curtis and Reiner section 15), but it doesn't seem to give an answer to the following question: 
Are there other "classes" of groups such that every irreducible representation is realizable over $\mathbb{Q}$?
Take "classes" to mean whatever you think appropriate (so long as it doesn't mean the collection of all groups with only rational irreps).

REPLY [9 votes]: Finite groups all of whose ordinary complex representations have rational-valued characters are sometimes called Q-groups (and sometimes called rational groups).  There is a monograph by Denis Kletzing (Structure and Representations of Q-Groups, Springer Lecture Notes in Mathematics 1084, 1984) which might be of interest.  Symmetric groups are Q-groups, as you mention, as are Weyl groups.  I can't remember if Kletzing constructs any other infinite families although I do remember that $D_n$ is a Q-group only when $n=1,2,3,4$ or $6$.  It's also worth mentioning that homomorphic images and direct products of Q-groups are also Q-groups.