Can one obtain a classification of 2-groups with center of index 4, analogous to the classification of 2-groups with derived subgroup of index 4?
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2$\begingroup$ Why are you particularly interested in 2-groups whose center has index 4 -- i.e. why not e.g. index 2 or 8, or 3-groups with center of index 9, etc., etc. -- and what type of answers are you looking for? $\endgroup$– Stefan Kohl ♦Aug 3, 2014 at 9:43
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$\begingroup$ @StefanKohl: Just to nitpick, it's very easy to characterize all groups with center of index 2: they do not exist.... $\endgroup$– Arturo MagidinAug 3, 2014 at 22:08
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$\begingroup$ @ArturoMagidin: Of course. -- But isn't the trivial characterization a nice characterization? $\endgroup$– Stefan Kohl ♦Aug 3, 2014 at 22:37
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$\begingroup$ A natural framework would be to classify finite groups with center of index $p^2$ (in a sense these, among $p$-groups, are the non-abelian groups with center as large as possible), and such a group being direct product of an abelian group and a $p$-group, we can stick to $p$-groups. Typically for such classifications $p=2$ should be a bit exceptional. $\endgroup$– YCorMar 7, 2020 at 8:25
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1 Answer
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These are the nonabelian central extensions of the Klein four-group by an abelian 2-group.
answered Aug 3, 2014 at 9:45
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1$\begingroup$ Derek: Note that I wrote "central extension". Dihedral 2-groups of order greater than 8 cannot be central extensions of the Klein four-group. $\endgroup$ Aug 3, 2014 at 15:40
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$\begingroup$ @DerekHolt: Can you explain a little more about this equivalent description? (This question came up on math.stackexchange.com/questions/2686286/…) $\endgroup$– verretMar 11, 2018 at 23:46
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$\begingroup$ @Verret It's not correct. I'll delete that comment. $\endgroup$ Mar 12, 2018 at 8:05