3. Determine the largest positive integer $N$, such that there exists a $6 \times N$ table $T$ satisfying:
(1) Each column is a permutation of $1,2, \cdots, 6$;
(2) For any two columns $i \neq j$, there exists a row $r(r \in$ $\{1,2, \cdots, 6\}$ ) such that $t_{r i}=t_{r j}$;
(3) For any two columns $i \neq j$, there exists a row $s(s \in$ $\{1,2, \cdots, 6\})$ such that $t_{s i} \neq t_{s j}$.
3. The largest positive integer $N=120$.

First, from any two columns being different, we know $N \leqslant 6$ !.
Also, from any two columns having at least one row with the same number, we know that among the six permutations $\left(a_{1}, a_{2}, \cdots, a_{6}\right),\left(a_{2}, \cdots, a_{6}, a_{1}\right), \cdots$, $\left(a_{6}, a_{1}, \cdots, a_{5}\right)$, at most one belongs to the columns of $T$. Thus, $N \leqslant \frac{6!}{6}=120$.
Furthermore, when $T$ is a $6 \times 120$ table where the first row of each column is 1 and the rest of each column is a permutation of $2 \sim 6$, it satisfies the conditions.
Therefore, $N_{\max }=120$.