= Solution
The eight-dimensional simple module is projective, so its <tensor product of group representations> with the natural three-dimensional module is projective. Its Brauer character is
$$
\phi_8\phi_3=(24,0,\alpha,\overline\alpha).
$$
From part (iii),
$$
\Phi_3+\Phi_8
=(24,0,\alpha,\overline\alpha).
$$
The duality of simple and projective Brauer characters makes the decomposition multiplicities unique. Hence, writing $P_3$ and $P_8$ for the corresponding projective covers,
$$
\boxed{8\otimes3\cong P_3\oplus P_8.}
$$
This completes the explicit <2-modular representation theory of GL3 of F2> calculation.
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