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I was solving a problem and in the middle, I came across this. I will be really grateful for any help here. In the following, we fix the integer $s\ge 2$.

Let $U\le \operatorname{GL}(2^s,\mathbb Z)$ be a finite $2$-group. Let $N=N_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ and $C=C_{\operatorname{GL}(2^s,\mathbb Z)}(U)$ be the normaliser and centraliser respectively. Suppose $N$ is acting on a set $S$, in particular $S\le \mathbb Z^{2^s}$ and action of $N$ is defined via "right multiplication".

My question is:

If we know the orbits in $S$ under the action $C$, then is there any result (or any method) which describes the orbits under $N$? Does the famous $N/C$ Theorem help in any way?

Any reference or hint will be greatly appreciated. Thanks in advance.

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    $\begingroup$ A start is provided by noting that $N$ permutes the $C$-orbits in a length preserving manner. $\endgroup$ Mar 9, 2020 at 22:58
  • $\begingroup$ Thanks. It will be very helpful if you kindly give some more hints and any reference please. Thanks again. $\endgroup$
    – usermath
    Mar 9, 2020 at 22:59
  • $\begingroup$ As well as the $C$-orbits, you need to understand how these are permuted by the group $N/C$. This would be explained by understanding the treatment of finite permutation groups in a group theory or algebra text. $\endgroup$ Mar 11, 2020 at 12:31

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