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gr.group idea – The centralizer and normalizer of merchandise of (whirl(n) $instances dots$) in U(m)
$DeclareMathOperatorSU{SU}DeclareMathOperatorU{U}DeclareMathOperatorwhirl{whirl}$
assume the whirl group $whirl(n)$ and the unitary group $U(16)$.
Below I specify a specfic path to embed $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16)$:

First we will simply embed the whirl group $whirl(10)subset U(16)$. Here we prefe the $mathbf{16}$dimensional spinor illustration of $whirl(10)$ to breathe too the $mathbf{16}$dimensional elementary illustration of $U(16)$. Thus, the info for the illustration:
$mathbf{16}$ in $whirl(10)$ as $mathbf{16}$ in $U(16)$
offers us an motion of $whirl(10)$ and $U(16)$ on the sophisticated vector area $mathbb{C}^{16}$. 
Then, we will simply embed $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset whirl(10) subset U(16)$. The mod $mathbb{Z}_2$ issue signifies that the main focus $Z(whirl(6))=Z(SU(4))=mathbb{Z}_2$ is recognized with the diagonal $mathbb{Z}_2$ subgroup of the main focus $Z(whirl(4))=Z(SU(2)instances SU(2))=mathbb{Z}_2 oplus mathbb{Z}_2$.
Here we denote:

the irreducible spinor illustration of $whirl(6)$ as ${mathbf 4}$
(which is just too the basic illustration of $SU(4)$). 
the complicatedconjugated illustration (of ${mathbf 4}$) as $overline{mathbf 4}$.

too $whirl(4)=SU(2)instances SU(2)$, the place we lookout the basic illustration of every SU(2).
Then, the info for the next illustration offers us an motion of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ on $mathbb{C}^{16}$, which additional offers an embedding of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ into $U(16)$:
$$textual content{$({mathbf 4},{mathbf 2},{mathbf 1}) oplus (overline{mathbf 4},
{mathbf 1}, {mathbf 2})$ in $(whirl(6)instances whirl(4))/mathbb{Z}_2$ as a decomposition of $mathbf{16}$ in $U(16)$}.$$
Question
1). Then my query is in regards to the centralizer and normalizer of this $$(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16),$$
which actually is determined by the embedding that I offered above. My distrust is that apart from some $U(1)$ components, there may breathe extra discrete finite teams.
2). Another refined loom is that given the an identical clique of illustration $({mathbf 4},{mathbf 2},{mathbf 1}) oplus (overline{mathbf 4}, {mathbf 1}, {mathbf 2})$,
it isn’t solely a illustration of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ however too a illustration of $(whirl(6)instances whirl(4))$. Thus we will query too the centralizer and normalizer of this embedding $$(whirl(6)instances whirl(4)) subset U(16)$$
apart from the $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16).$
assassinate we maintain the an identical or completely different solutions of centralizer and normalizer for the illustration 1) versus 2)?
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