if $ N $ is a regular subgroup of $ {\text{Perm}}(G) $ normalised by $\lambda(G) $ then the Hopf algebra giving the Hopf-Galois structure corresponding to $ N $ is $ L[N]^{G} $, the fixed ring of the group algebra $ L[N] $, where $ G $ acts on $ L[N] $ by acting on $ L $ as Galois automorphisms and on $ N $ by $ \,^{g} \eta = \lambda(g)n\lambda(g^{-1})$ for all $ \eta  \in N $, $ g \in G $. For a Hopf algebra $H=L[N]^G$ giving a Hopf Galois structure on $L/F$, we refer to $N$ as the *underlying group* of $H$ and its isomorphism class as the *type* of $H$, or the structure given by $ H $.

Let $ \rho : G \hookrightarrow {\text{Perm}}(G) $ be the right regular representation. Suppose $g,h\in G$ and $x\in G$. Then $\lambda(g)\rho(h)[x]=gxh^{-1}=\rho(h)[gx]=\rho(h)\lambda(g)[x]$. That is: $\lambda(g)\rho(h)=\rho(h)\lambda(g)$ for all $g,h\in G$. Thus the action of $G$ on $\rho(G)$ is trivial, and so the Hopf algebra $L[\rho(G)]^G$ is in fact the group algebra $F[G]$ as in the original discussion. The Hopf Galois structure for which $N=\rho(G)$ is the underlying group is called the *classical* structure.

It is clear that the action 