Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le j_i\le l_i$ for $i=1,...,n$.
This requirement is stronger than demanding a generating set because the product could be $x^2 y^3 x^2$, for example, which is not allowed in the restricted version. It is clear that the product of the orders of the $x_i$ must be at least the order of the group. It is also clear that $l_i$ must be smaller than the order of the element $x_i$.
Example : For the permutations $x=(1347)(2568)$ and $y=(1246)(3875)$, the set $[e,x,x^2,x^3,y,xy,x^2y,x^3y]$ forms the quaternion group $Q_8$.
Does a set with the required property exist for every finite permutation group $G$ ?
How can I find the smallest possible set and the exponents ? (The example shows that the limit for the exponent for some element need not be equal to the order of the element)
GAP allows to check whether a given set forms the required group, but the brute-force-method will soon become infeasible.