= Solution
Only the classes $1,3,7A,7B$ are 2-regular. The trivial module and the natural three-dimensional module are simple; the dual natural module gives the conjugate three-dimensional character. Restricting the ordinary characters to the odd-order classes and using $\alpha+\overline\alpha=-1$ produces the fourth simple character of degree eight. The <Brauer character> table is
$$
\boxed{
\begin{array}{c|rrrr}
&1&3&7A&7B\\\hline
\phi_1&1&1&1&1\\
\phi_3&3&0&\alpha&\overline\alpha\\
\phi_{\overline3}&3&0&\overline\alpha&\alpha\\
\phi_8&8&-1&1&1
\end{array}}
$$
The ordinary degree-eight character restricts exactly to $\phi_8$. Since its degree contains the full 2-part $8$ of $|G|$, this simple module is projective and its singleton block has defect zero. Thus
$$
\boxed{\phi_8\text{ lies in the unique defect-zero block};
\quad\phi_1,\phi_3,\phi_{\overline3}\text{ lie in the principal block}.}
$$
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