Background
Imagine you have a tuple of integers, say $t=(1, 2, 0, 3, 1, 5, 1)$. Then you can order the list with your favourite sorting algorithm and obtain $t'=(5, 3, 2, 1, 1, 1, 0)$. An alternative way of finding this element is
$$ t'=\max\{ gt : g\in S_7 \} $$ for the symmetric group $S_7$, and where we mean max with respect to lexicographical order. Of course listing all possible permutations is inefficient, which is why we invented sorting.
Actual Question
Now take a subgroup $G\trianglelefteq S_7$, which acts on tuples of integers; for our example take the group generated by the two cycles $(12)(34)$ and $(567)$. How do we find $$ t''=\max\{ gt : g\in G \} $$ without always calculating the entire orbit of $t$ under $G$?
What I Have
I suspect one can use a strong generating set for this; but I can't think of a way to do it.
An interesting reference I found is arXiv:1211.6261 where, in listing 1, the authors explain how to check whether a tuple is already the maximum within the orbit, with worst-case runtime being the size of the orbit. Since I'm looking for a sorting algorithm I would expect a worst-case runtime to be $\Omega( n \log n)$ for a tuple of length $n$; so generating the entire orbit is potentially exponentially worse.
Thanks! - J