I am looking for any results on Schur indices over $\mathbb{Q}$ for 2-groups. By a theorem of Roquette (corollary 10.14 in Isaacs) these numbers are at most 2. I am interested in 2-groups for which the maximum value is attained. That is, I am interested in 2-groups having irreducible characters of Schur index 2 over $\mathbb{Q}$. A class of examples is the generalized quaternion groups. Are there quaternion-free examples?
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1$\begingroup$ It may depend on the sense in which you use the term "quaternion-free". If you intend this to mean "with no quaternion section of order $8$," I would guess the answer would be "no", but maybe you mean it in a different sense. $\endgroup$– Geoff RobinsonCommented Feb 5, 2020 at 12:06
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$\begingroup$ This is a nice database for looking for finite groups with particular properties people.maths.bris.ac.uk/~matyd/GroupNames $\endgroup$– Henri JohnstonCommented Feb 5, 2020 at 12:48
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$\begingroup$ @GeoffRobinson that is exactly the sense of “quaternion-free” I had in mind. I think the answer is “no” as well, and it seemed likely to me that a result like that should be in the literature somewhere. $\endgroup$– John McHughCommented Feb 5, 2020 at 13:40
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