Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would appreciate advice into what specific areas/theorems I should learn that apply to this particular question.

**Here is the problem**: I have a group $G$ acting on a very big vector space $V$, which is over $\mathbb{F}_2$. I want to find the orbits (specifically, representatives) of the action of $G$ on $V$. I could just go through each element and then sort for redundancies, but $V$ is too big for that to be feasible. My hope is to be able to use the basis of $V$, which is small, instead. I * can* compute the image of each basis vector under each linear transformation induced by $G$. More specifically, I can determine the coefficients of each image too.

*Is there a way to understand and calculate specific orbits of the entire vector space using information about the basis specifically?*

I figure there might be a very general way of understanding the orbits of group representations—that is, of elements of the vector space that are "distinct $\mod G$"—but I do not know enough about character theory, homology, etc. to see the direction to go. If anyone could recommend to me some specific readings that are relevant to problems like this, I would very much appreciate it. Thanks.

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