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Can anyone help in writing code to find the list of idempotent and primitive elements of a group algebra?

The examples goes like this. Let $p$ be an odd prime such that $\bar2$ generates $U(Z_{p^2})$ and let $G =(\text{ideal generated by }a) ∗ (\text{ideal generated by }b)$ an abelian group, with $o(a)=p^2$ and $o(b)=p$.

Then $F_2G$ has four inequivalent minimal codes, namely, the ones generated by the idempotents:

$$e_0 = \hat{G}$$ $$e_1=\hat{b}−\widehat{(\text{ideal generated by }a)∗ (\text{ideal generated by }b)}$$ $$e_2=\widehat{a−G}$$ $$e_3=\widehat{(\text{ideal generated by }a_p)∗ (\text{ideal generated by }b)}−\hat{G}$$