let $ F_{i} $ be the fixed field of ${\text{Stab}}_{\Omega}(\chi_i)$; then $$H \cong \prod_{i=1}^{s} F_{i} \mbox{ as $ F $-algebras.}$$ A result of Böltje and Bley [@BLE1999 Lemma 2.2] shows how one may construct an $ F $-basis of $ L[N]^G $ corresponding to this decomposition: we have $ L[N]^{G} =  F^{\text{alg}}[N]^{\Omega} $, and the group algebra $ F^{\text{alg}}[N] $ has a basis of mutually orthogonal idempotents, each corresponding to an element of $ \widehat{N} $. The action of $ \Omega $ on $ F^{\text{alg}}[N] $ permutes these idempotents, and by forming $ \Omega $-invariant linear combinations we obtain an $ F $-basis of $ L[N]^{G} $ corresponding to the decomposition above.

If $ H=L[N]^G $ is a Hopf algebra whose underlying group $ N $ is isomorphic to $C_2\times C_2\times C_2$ then the values of the characters of $ N $ lie in $ F $, so the action of $ \Omega $ on $ \widehat{N} $ factors through $ G $. Using this observation we have:

Let $ E_{s,t} $ be defined as in . Then we have $$L[E_{s,t}]^G\cong F^4\times K  \mbox{ as $ F $-algebras.}$$

The dual group $\widehat{E}_{s,t}$ is generated by three characters: $$\begin{aligned}
&\chi_1:\left\{
\begin{array}{rcl}
