Efficient method to determine the order of a permutation in $S_n$ Instead of trying multiplication again and again until I get $(1)(2)(3)(4)(5)(6)(7),$ is there an efficient, logical method to compute order of $(157)(134)(12)$ of $S_{10}$?
Is there some relation to determine it?
Also, I am aware there is some relation with order of group, but don't remember exactly. I would be grateful if you could help.
Thank you.

We can easily compute the order of a permutation which is written as the product of disjoint cycles.  
So first we need to "multiply out" (compose) the permutation expressed as the product of non-disjoint cycles you posted. Let's call your permutation $\alpha$:
$$\alpha = (\color{blue}{\bf 1}57)(\color{blue}{\bf 1}34)(\color{blue}{\bf 1}2) \in S_{10}$$
I've highlighted in blue why $\alpha$ fails to be the product of disjoint cycles: $1$ appears in each of the three factor-cycles, and in each, is permuted to different values: $1\to 5,\;1\to 3, \;1\to 2$
So, when we compose these cycles that are each a factor of $\alpha$, we obtain the one-cycle that caveman provides:
$$\alpha = (157)(134)(12) = (6)(1\,5\,7\,3\,4\,2)(8)(9)(10) = (1\,5\,7\,3\,4\,2).$$
Now we can address the order of a permutation:
We define the length of a cycle to be the number of elements in the cycle. 
We define the $|$ order $|$ of a permutation written as the product of disjoint cycles to be the least common multiple $(\operatorname{lcm})$ of the lengths of those cycles. So for $\rho \in S_n$, written as the product of $k$ disjoint cycles:
$$\operatorname{order}(\rho) = |\rho| = |\sigma_1\sigma_2\cdots\sigma_k| = \operatorname{lcm}(|\sigma_1|, |\sigma_2|, \ldots , |\sigma_k|)$$
Since $\alpha =(1\,5\,7\,3\,4\,2)$ is a "one-cycle", its order is equal to its length, which is $6$.
If we had the permutation, e.g.: $$\beta = (1\,2\,3)(4\,5)(6\,7\,8),$$ then the order of $\beta$ would be $\operatorname{lcm}(3, 2, 3) = 3\cdot 2 = 6$.
