Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}\_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}\_{2^k}(\mathbb{F}\_2)$, and we computed it - it was not too easy. It's an abelian group, and so we were able to compute the decomposition of the quotient of the centralizer over the subgroup generated by $C$ into the sum of cyclic subgroups, as follows. 
$$
\bigoplus\_{i=2}^k (\mathbb{Z}/\mathbb{Z}\_{2^{k+1-i}})^{2^{i-2}}.
$$


We wonder if this was already done. (We also did this for more general case of other primes, not only 2, formulas are similar).


Update: see [centralizer of a n-cyclic permutation matrix over F\_2 in GL(n,2)](https://mathoverflow.net/questions/140280/centralizer-of-a-n-cyclic-permutation-matrix-over-f-2-in-gln-2) for a follow-up question.



centralizer of the order 2^k cyclic permutation matrix over F_2
the details of our computation can now be found in <http://arxiv.org/abs/1405.0113>


