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Let $G$ be a finite group. A (complex) character $\chi$ of $G$ is said to be a virtual permutation character if it can be expressed as a $\mathbb{Z}$-linear combination of characters induced from the trivial characters of subgroups of $G$, i.e., $\chi = \sum_{H \leq G} n_H \mathrm{ind}^{G}_{H} 1_H$ for some $n_H \in \mathbb{Z}$.

My question is: which finite groups have the property that every character is a virtual permutation character? Is there a classification of such groups? If so, can you provide a reference?

One well-known example is the symmetric group $S_n$. Moreover, it is clear that the class of groups we're looking for is closed under direct products. But are there other examples?

Note that every virtual permutation character is rational-valued, but the converse is false. For example, the every character of the quaternion group of order $8$ is rational-valued, but the unique non-linear character is not a virtual permutation character. Moreover, non-trivial Schur induces are not the only obstacle: even if the representation attached to $\chi$ takes values in matrices over $\mathbb{Q}$, the character $\chi$ need not be a virtual permutation character. These obstacles are the topic of the article "Rational representations and permutation representations of finite groups" Math. Ann. 364, Issue 1 (2016), 539-558 by Alex Bartel and Tim Dokchitser (see http://arxiv.org/abs/1405.6616).

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  • $\begingroup$ A quick search by computer algebra found some further examples: the dihedral group of order $8$, and the split extension of $\langle x, y \rangle \cong C_3 \times C_3$ by a fixed-point-free automorphism $t$ of order $2$, acting as $x^t = x^2$, $y^t = y^2$. $\endgroup$ – Mark Wildon Jul 28 '16 at 20:59

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