# Orbit calculation for normaliser when orbits under centraliser action is known

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.

• A start is provided by noting that $N$ permutes the $C$-orbits in a length preserving manner. Mar 9, 2020 at 22:58
• Thanks. It will be very helpful if you kindly give some more hints and any reference please. Thanks again. Mar 9, 2020 at 22:59
• 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. Mar 11, 2020 at 12:31