Example 46 Let $a_{i}, b_{i}, c_{i}(i=1,2, \cdots, 6)$ all be permutations of $1,2, \cdots, 6$, find the minimum value of $\sum_{i=1}^{6} a_{i} b_{i} c_{i}$.
Let $S=\sum_{i=1}^{i} a_{i} b_{i} c_{i}$, then
$$
S \geqslant 6 \sqrt[6]{\prod_{i}^{6} a_{i} b_{i} c_{i}}=6 \sqrt[6]{(6!)^{3}}=72 \sqrt{5}>160 .
$$

Next, we prove $S \geqslant 162$. Since the geometric mean of $a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}, \cdots, a_{6} b_{6} c_{6}$ is $12 \sqrt{5}$, and $26161 .
\end{aligned}
$$

Furthermore, when the three sequences are arranged as follows,
$$
\begin{array}{llllll}
\left\{a_{i}\right\}: 1 & 5 & 6 & 2 & 3 & 4 \\
\left\{b_{i}\right\}: 5 & 6 & 1 & 3 & 4 & 2 \\
\left\{c_{i}\right\}: 6 & 1 & 5 & 4 & 2 & 3 \\
S=3 \times 30+3 \times 24=162 .
\end{array}
$$

Therefore, the minimum value of $S$ is 162.