Longest Rubik's cube algorithm - maximization problem

Question

For a given Rubik's cube algorithm $A$ let $\mathfrak C(A)$ be the number of times, we have to repeat algorithm $A$ to get back to where we've started. For example if $A=RUR'U'$ then $\mathfrak C(A)=6$ The question is:
What is the greatest value of $\mathfrak C(A)$, we can achieve?
This question is equivalent to following maximization problem:
Maximize $LCM(a_1,a_2,...a_i,b_1,b_2,...b_j)$ under the following conditions: $$~a_1,a_2,...a_i,b_1,b_2,...,b_j \in \mathbb N_+~$$ $$a_1+a_2+...+a_i=8$$ $$b_1+b_2+...+b_j=12$$ $$2 \ | \ (a_1+a_2+...+a_i+b_1+b_2+...+b_j)$$ Where LCM is Least Common Multiple function.
Thanks for all the help.

Answer 1

Wikipedia's article on the Rubik's cube group says that the largest order of any element in the group is 1260. For instance, $RU^2D'BD'$ is one such move.