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Consider the group action of $S_5$ on $(\mathbb{Z}/5\mathbb{Z})^6$ given by $$(12345)\colon (a,b,c,d,e,f)\mapsto(b,c,d,e,a,f)$$ $$(12)\colon (a,b,c,d,e,f)\mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$ Do we know how to compute the number of orbits of the action? Do we know how to calculate the length of each orbit? I am interested in what possible stabilizer subgroup groups would appear.

To me it appears this problem is this is too large to calculate by hand, and I hope someone could help to solve it by computer programming.

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For computing the number of orbits I suggest Burnside's lemma: https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

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