I have a bit string, $X$, of $n$ bits. I am applying an operation, $g$, which involves a permutation of the bits and then flipping the first $k$ bits.
I have a proof that the $g$ generates a group because $$g(gX)=ggX$$ and $$g^rX=X$$ where $r$ is the order of a permutation applied to an enlarged representation, $Y$, of $X$. Where $Y$ is of length $2n$ and $Y[2i-1] = X[i]$ and $Y[2i] = \lnot X[i]$. This results in changing all the bit flips to permutations.
However, $r$ is in general much larger than the orbit of $X$ under $g$.
Are there any techniques to find the size of this orbit other than computing them all?
Can I decompose g into a composition of a permutation followed by another group operation and use that to find the orbit?