I am puzzled by the following structure (assume it is a group), called it $G$, contains and satisfies the following ingredients (in particular we can focus on the case that $d=2$). The $G$ contains generators including
$\mathbb{Z}/2 \equiv \mathbb{Z}_2^A$
$SO(d)$
$\mathbb{Z}/4 \equiv \mathbb{Z}_4^B$
The subindices $A$ and $B$ are to specify they are different groups. The group operations of $\mathbb{Z}_2^A$, $SO(d)$, and $\mathbb{Z}_4^B$ satisfy:
- (1) $\mathbb{Z}_2^A$ and $SO(d)$ do not commute:
They obey $\mathbb{Z}_2^A \rtimes SO(d) = Spin(d)$, where $\frac{Spin(d)}{\mathbb{Z}_2^A}=SO(d)$. Namely, we have $ 1 \to \mathbb{Z}_2^A \to Spin(d) \to SO(d) \to 1$.
- (2) $SO(d)$ and $\mathbb{Z}_4^B$ do not commute:
They obey $SO(d) \rtimes \mathbb{Z}_4^B =E(d)$, in fact $E(d)$ obeys that the quotien group $\frac{E(d)}{SO(d)}=\mathbb{Z}_4^B$ and $\frac{E(d)}{\mathbb{Z}_2^B}=O(d)$ and $\frac{\mathbb{Z}_4^B}{\mathbb{Z}_2^B}=\mathbb{Z}_2'$ where $\mathbb{Z}_2'$ is an another mod 2 abelian group. Namely, we have three short exact sequences $ 1 \to SO(d) \to E(d) \to \mathbb{Z}_4^B \to 1$, and $ 1 \to \mathbb{Z}_2^B \to E(d) \to O(d) \to 1$, and $ 1 \to \mathbb{Z}_2^B \to \mathbb{Z}_4^B \to \mathbb{Z}_2' \to 1$.
We can also write $E(d)$ as $$E(d) =\{ (M, j) \in (O(d), \mathbb{Z}_4^B) \; \vert \; \det M = j^2\}$$ where $j \in \mathbb{Z}_4^B$ here can be written as $\{1,i,-1,-i\}$ with $i^4=+1.$
- (3) $\mathbb{Z}_4^B$ and $\mathbb{Z}_2^A$ do not commute:
They obey $\mathbb{Z}_4^B \rtimes \mathbb{Z}_2^A = D_8$ is a dihedral group of order 8. Namely, we have $ 1 \to \mathbb{Z}_4^B \to D_8 \to \mathbb{Z}_2^A \to 1$.
In summary, $G$ contain three group generators $$ \mathbb{Z}_2^A, SO(d),\mathbb{Z}_4^B $$ and group operations defined by $$ \mathbb{Z}_2^A \rtimes SO(d) = Spin(d),\;$$ $$ SO(d) \rtimes \mathbb{Z}_4^B =E(d),\;$$ $$\mathbb{Z}_4^B \rtimes \mathbb{Z}_2^A = D_8. $$
The curious part is that the semi direct products $\rtimes$ are given in a way that $ \mathbb{Z}_2^A, SO(d),\mathbb{Z}_4^B$ can be a normal subgroup repsectively in (1), (2), (3); but $ \mathbb{Z}_2^A, SO(d),\mathbb{Z}_4^B$ can be a quotient subgroup respectively in (3), (1), (2). So their group operations are cyclic in some way.
So what are the precise constructions/descriptions of the group $G$? Does this generate some familiar groups?
p.s. any comments/partial answers/Refs are welcome!