The number of Hopf-Galois structures on a quaternionic extension

\[table\_count\]

We now determine the regular subgroups of $ {\text{Perm}}(G) $ corresponding to these Hopf-Galois structures. We start with the subgroups corresponding to the Hopf-Galois structures of type $C_2\times C_2\times C_2$.

\[lemma\_Est\] Let $s,t\in\{\sigma,\tau\} $ with $ s\neq t$ and let $E_{s,t}$ be generated by $\lambda(s)\rho(t),\lambda(s^2)$, and $\lambda(t)\rho(st)$. Then $E_{s,t}$ is a regular subgroup of ${\text{Perm}}(G)$ that is normalized by $\lambda(G)$ and isomorphic to $C_2\times C_2\times C_2$. The groups $E_{\sigma,\tau}$ and $ E_{\tau,\sigma} $ are distinct, and are the underlying groups of the 2 Hopf-Galois structures of type $C_2\times C_2\times C_2$ on $ L/F $.

The elements of $E_{s,t}$ are $$1,\,\,\lambda(s^2),\,\,\lambda(s)\rho(t),\,\,\lambda(s^{-1})\rho(t),\,\,\lambda(t)\rho(st),\,\,\lambda(t^{-1})\rho(st),\,\,\lambda(st)\rho(s),\,\,\lambda((st)^{-1})\rho(s).$$ All of the non-identity elements above have order 2, so $E_{s,t}$ is isomorphic to $C_2\times C_2\times C_2$. It is clear that $ E_{s,t} \subset {\text{Perm}}(G) $ and $E_{s,t}\cdot1_{G}=G$; hence $E_{s,t}$ is a regular subgroup of ${\text{Perm}}(G)$. To show that $E_{s,t}$ is normalized by $\lambda(G)$, it is sufficient to show that it is normalized by $ \lambda(s) $ and $ \lambda(t) $. Using the fact that $\lambda(G)$ and $\rho(G)$ commute inside $ {\text{Perm}}(G) $ we have for example $$\begin{aligned}
\,^{s} \lambda(s)\rho(t) & = \lambda(sss^{-1})\rho(t) = \lambda(s)\rho(t) \\
\,^{t} \lambda(s)\rho(t) & = 