TITLE: Induced trivial representation
QUESTION [6 upvotes]: Let $G$ be a finite group, and $H$ be a subgroup of $G$.
If I am not mistaken, we have
$$Ind_H^G \,\,1_H=1_G+\sum_{\chi\neq 1} n_\chi \chi$$
where $\chi$ are irreducible representations.

My question is, what representations are these?

I guess is should be a proper subset (if $H$ is a proper subgroup) of the irreducible representations of $G$, but I'm not too sure of that either.
And if that's correct, how do I know which $\chi$ occur in the decomposition above.

REPLY [3 votes]: There is a nice interpretation of $1_{H}^G$ in terms of permutation characters. Let $X$ be a set on which $G$ acts. Then the character $\chi$ of the permutation action is given by the following formula:
$\chi(g)=\#\{$fixed points of $g$ on $X$$\}$. 
Now, if $G$ acts transitively on $X$ and $H=H_x$ is the stabilizer subgroup of $G$ on one of the points $x \in X$, then $\chi=1_{H}^G$.