This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups.

More precisely, given discrete groups below (a), (b), (c):

(a) dihedral group of order 8: $D_4$,

(b) quaternion group of order 8: $Q_8$, and

(c) elementary group of order 8: $(\mathbb{Z}_2)^3$

can you find out a list of positive integer $n$ such that these three discrete groups respectively are being contained by:

(1) $\mathrm{SO}(n)$,

(2) $\mathrm{SU}(n)$,

(3) $\mathrm{Spin}(n)$,

(4) $\mathrm{SO}(n)\times\mathrm{SO}(n)$,

(5) $\mathrm{SU}(n)\times\mathrm{SU}(n)$.

Thank you very much!

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    $\begingroup$ You can more or less answer this question using the representation theory of finite groups. Are you familiar with it? $\endgroup$ – Qiaochu Yuan May 23 '13 at 5:23
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    $\begingroup$ @Qiaochu Yuan, can you illuminate this: math.stackexchange.com/questions/2047197, I saw your nice blog post on SU(2) and quaternion. $\endgroup$ – miss-tery Dec 6 '16 at 23:53
  • $\begingroup$ @Qiaochu Yuan, It seems to me that my questions (2) and (3) are impossible, thus negative attempt--- except the only example I gave: SU(2)/Z2=SO(3)SU(2)/Z2=SO(3)? Perhaps my question (1) is also negative: math.stackexchange.com/questions/2047197 $\endgroup$ – miss-tery Dec 6 '16 at 23:54

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