(Please skip to the last paragraph if you are interested in just the question)
I wish to compute the generators of the ring of invariants for a symmetric group acting on a polynomial ring using a computer software. For concreteness,consider $S_5$. Then, there are 7 irreducible representations of $S_5$. For each of these, I wish to do the above computations. The method being followed by me is as follows on SageMath.
For each representation, I write down the matrices corresponding to the action of (1,2) and (1,2,3,4,5) on the polynomial ring. As these 2 permutations generate $S_5$, the matrices will generate the required matrix group. Then, using the invariant_generators() function of SageMath, I perform the required computation.
This method worked successfully for the linear representations, standard representations and the 5-dimensional representations. But, for the 6-dimensional representation, the software is taking too much time for computation. I left my computer on overnight, and still, my desired computations weren't complete by morning.
So, my problem is, what is the fastest method in SageMath for computing the generators of the ring of invariants of the action of a finite group on a polynomial ring? Also, is there any other software, available for free usage (unlike MAGMA), which will work better than SageMath for the given computation?