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A finite crossed module is a 4-tuple $$(G_1,G_2,\delta: G_2 \to G_1, \alpha: G_1 \to Aut(G_2))$$

satisfying certain compatible conditions, where the $G_i$ are finite groups and the maps are group homomorphisms.

  1. Given two arbitrary finite groups $G_1, G_2$, are all finite crossed modules with the same first two entries known?
  2. Given a group homomorphism $\delta$ from $G_2$ to $G_1$, can we describe the corresponding set of equivalent crossed modules from $H^3(K,A)$, where $$ A = ker(\delta), K = cok(\delta)? $$

If unfortunately both cases are wildly unknown.. are there some situations in which the classification problem has been settled?

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  • $\begingroup$ You could look at HOMOTOPY 2-TYPES OF LOW ORDER GRAHAM ELLIS AND LE VAN LUYEN in the J Experimental Math, $\endgroup$
    – Ronnie Brown
    Commented Jul 16, 2020 at 10:13
  • $\begingroup$ @RonnieBrown It's very helpful. With both groups fixed, up to quasi-isomorphism the structure only depends on the Postnikov invariant. How about "up to isomorphism"? In section 3 the author implemented an algorithm, which seems to be brute forcing all possibilities.. To summarize.. is there a more conceptual way to classify finite Cat1Group/CrossedModules structures up to isomorphism? $\endgroup$
    – Student
    Commented Jul 16, 2020 at 13:24

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As you have discovered, the method in the GAP package HAP constructs all cat1-groups G->G, and provides a library of all cat1-group structures on groups of order up to 255 (with some exceptions). The XMod package also has an operation AllCat1GroupsUpToIsomorphism(G). But it is not clear how to implement an operation AllXModsUpToIsomorphism(G1,G2) - AllHomomorphisms(G1,G2) is available, in GAP, but then all possible actions would be required. Would such a function be useful? For a crossed module X0, the XMod package provides an operation KernelCokernelXMod(X0) which constructs your A->K. None of this, of course, involves a conceptual approach.

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  • $\begingroup$ The questioner could also find the following helpful on the use of crossed modules. 1. "Groupoids and crossed objects in algebraic topology"Homology, Homotopy Applications, 1(1999) 1-78, and the book "Nonabelian Algebrqic Topology, both downloadable from my wen site. The uses suggest the kind of classification one seeks. . $\endgroup$
    – Ronnie Brown
    Commented Jul 23, 2020 at 14:15

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