Given an indecomposable $kG$-module $M$, can MAGMA compute a minimal projective cover of $M$?

Question

I'd like to ask the following MAGMA question:

Let $G$ be a finite group and let $k$ be a finite field, such that char$(k)$ divides $|G|$.

Given an indecomposable $kG$-module $M$, can MAGMA compute a minimal projective cover of $M$?

If yes, what is the appropriate command?

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I think "ProjectiveCover" did not always compute a minimal projective cover...I might do some calculations via the projective cover of the top, but I was wondering, whether there is a predefined command.

Answer 1

I don't think there is any command for doing this other than $\mathtt{ProjectiveCover}$ (which I wrote the code for). I am slightly surprised that it does not always give the minimal projective cover. Can you give me an example where it does not?

For a give module module $M$, $\mathtt{ProjectiveCover}$ computes the Jacobson Radical $J$ of $M$, and then writes $M/J$ as a direct sum $I_i \oplus \cdots \oplus I_r$ of irreducible modules $I_i$. It calculates the projective cover of $P$ as $P_i \oplus \cdots \oplus P_r$, where $P_i$ is the projective indecomposable module with quotient $I_i$.