$ on $ L/F $.

It is clear that $C_{s,t}$ is a subgroup of $ {\text{Perm}}(G) $ isomorphic to $C_8$. Moreover, we have $C_{s,t}\cdot 1_{G}=G$ since $\eta_{s,t}^k\cdot1_{G}=1_{G}$ if and only if $k \equiv 0 \pmod{8}$. Thus $C_{s,t}$ is a regular subgroup of ${\text{Perm}}(G)$. To show that $C_{s,t}$ is normalized by $\lambda(G)$, it is sufficient to show that it is normalized by $ \lambda(s) $ and $ \lambda(t) $. We have $$\begin{aligned}
\lambda(s)&\eta_{s,t}\lambda(s^{-1})\\
&=(1\,\,s\,\,s^2\,\,s^{-1})(t\,\,st\,\,t^{-1}\,\,(st)^{-1})(1\,\,s\,\,t\,\,(st)^{-1}\,\,s^2\,\,s^{-1}\,\,t^{-1}\,\,st)(1\,\,s^{-1}\,\,s^2\,\,s)(t\,\,(st)^{-1}\,\,t^{-1}\,\,st)\\
&=(1\,\,(st)^{-1}\,\,t^{-1}\,\,s\,\,s^2\,\,st\,\,t\,\,s^{-1})\\
&=\eta_{s,t}^3,\end{aligned}$$ and similarly, $\lambda(t)\eta_{s,t}\lambda(t^{-1})=\eta_{s,t}$. Therefore $C_{s,t}$ is normalized by $\lambda(G) $. It may be verified that each of the 6 choices of the pair $s,t$ gives a permutation that differs from all powers of those of the other choices. Hence, by , the groups $C_{s,t}$ are the underlying groups of the 6 Hopf Galois structures of type $C_8$.

Having found the abelian underlying groups of the corresponding Hopf Galois structures on our extension $L/F$ we now find the structures of quaternionic type which we saw earlier.

$\rho(G)$ and $\lambda(G)$ are the underlying groups of the the two Hopf-Galois structures of type $Q_8$.

As $G$ is non-abelian, $\rho(G) $ and $ \lambda(G)$ are distinct regular subgroups of $ {\text{Perm}}(G) $ normalized by $ \lambda(G) $. By , 