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I have the following problem. It is well known that $H^\ast(D_8,\mathbb{Z}/2)\cong \mathbb{F}_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). So we get that. $$ H^2(D_8,\mathbb{Z}/2)=\mathbb{F}_2\langle x^2, y^2,w\rangle. $$

I wanted to ask if there is any method/reference for describing extensions corresponding to particular cohomology classes. I am especially interested in classes $x^2+y^2+w$ and $x^2+w$.

Also, a broader question is - are there any general, or more general, methods of constructing extensions given a class in the second cohomology of a group? Any reference or help would be appreciated.

EDIT: $D_8$ here is the group of symmetries of a square.

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    $\begingroup$ For your second question I suppose it depends how the class is given. If you have an explicit 2-cocycle then the extension is easy to write down. In your example I suppose you need explicit cocycles for each of the generators $\endgroup$
    – Benjamin Steinberg
    Commented Feb 23, 2023 at 14:49
  • $\begingroup$ @BenjaminSteinberg I see, I firstly posted, then looked through the internet (not the best behaviour). Here is the method: ncatlab.org/nlab/show/… $\endgroup$
    – Igor Sikora
    Commented Feb 23, 2023 at 14:50
  • $\begingroup$ However, I'm still interested in the question about $D_8$. I suppose that the extensions are already well-studied, so if I can avoid computing cocycles by hand and look into some references, that would be great :) $\endgroup$
    – Igor Sikora
    Commented Feb 23, 2023 at 14:52
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    $\begingroup$ Could I just point out that there are two conventions for notation for the dihedral group, and you have not said which one you are using. $D_n$ can denote the dihedral group of order $n$ or of order$2n$. $\endgroup$
    – Derek Holt
    Commented Feb 23, 2023 at 16:07
  • $\begingroup$ @DerekHolt Thank you, that's fair point. Edited. $\endgroup$
    – Igor Sikora
    Commented Feb 23, 2023 at 17:08

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Here is a start for your specific questions. It is easy to see that under the inclusion $C_4 < D_8$, $x$ and $y$ both map to the 1 dimensional class and $w$ maps to the nonzero 2 dimensional class. So one concludes that a group $P$ of order 16 fitting into a central extension $$ C_2 \rightarrow P \rightarrow D_8$$ corresponds to an element of $H^2(D_8)$ that involves $w$ if and only if $P$ has a cyclic subgroup of order 8, as $C_8$ is the group fitting into the nontrivial central extension
$$ C_2 \rightarrow C_8 \rightarrow C_4.$$

Checking the wonderful website GroupNames (https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html) one sees that there are only three groups of order 16 that both have $D_8$ (called $D_4$ there) as a quotient and $C_8$ as a subgroup: $D_{16}$, $Q_{16}$, and $SD_{16}$.

I think that $w$ corresponds to $D_{16}$ since I think one gets the right cohomology ring. (See another fun website: https://users.fmi.uni-jena.de/cohomology/16web/index.html) Symmetry considerations make me then guess that $x^2+y^2+w$ corresponds to $Q_{16}$ and $x^2+w$ corresponds to $SD_{16}$.

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