# Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

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!

• You can more or less answer this question using the representation theory of finite groups. Are you familiar with it? – Qiaochu Yuan May 23 '13 at 5:23
• @Qiaochu Yuan, can you illuminate this: math.stackexchange.com/questions/2047197, I saw your nice blog post on SU(2) and quaternion. – miss-tery Dec 6 '16 at 23:53
• @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 – miss-tery Dec 6 '16 at 23:54