Solution
= Solution
There are $p^n$ possible translation vectors. An <invertible matrix> is specified by an ordered <basis> of $\mathbb F_p^n$. Its first column can be any nonzero vector, giving $p^n-1$ choices. After choosing $i$ independent columns, their span has $p^i$ elements, so the next column has $p^n-p^i$ choices. Therefore
$$
\boxed{|G|=p^n|\operatorname{GL}_n(\mathbb F_p)|
=p^n\prod_{i=0}^{n-1}(p^n-p^i).}
$$