Second cohomology group for trivial group action of direct product of D8 and Z2 on Z2

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This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group direct product of D8 and Z2 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and direct product of D8 and Z2 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E64.
Get more specific information about direct product of D8 and Z2 |Get more specific information about cyclic group:Z2|View other constructions whose value is elementary abelian group:E64

GAP implementation

Construction of the cohomology group

The cohomology group can be constucted using the GAP functions CyclicGroup, DirectProduct, TwoCohomology, TrivialGModule, GF.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));;
gap> A := TrivialGModule(G,GF(2));;
gap> T := TwoCohomology(G,A);
rec( group := <pc group of size 16 with 4 generators>,
  module := rec( field := GF(2), isMTXModule := true, dimension := 1,
      generators := [ <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2> ] ),
  collector := rec( relators := [ [ 0 ], [ [ 2, 1, 3, 1 ], [ 3, 1 ] ],
          [ [ 3, 1 ], [ 3, 1 ], 0 ], [ [ 4, 1 ], [ 4, 1 ], [ 4, 1 ], 0 ] ],
      orders := [ 2, 2, 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1, 3, 1 ] ],
      estack := [  ], pstack := [ 3, 5 ], cstack := [ 1, 1 ],
      mstack := [ 0, 0 ], list := [ 0, 0, 0, 0 ],
      module := [ <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2> ],
      mone := <an immutable 1x1 matrix over GF2>,
      mzero := <an immutable 1x1 matrix over GF2>, avoid := [  ],
      unavoidable := [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] ),
  cohom := <linear mapping by matrix, <vector space of dimension 7 over GF(
    2)> -> ( GF(2)^6 )>,
  presentation := rec( group := <free group on the generators
        [ f1, f2, f3, f4 ]>,
      relators := [ f1^2, f1^-1*f2*f1*f3^-1*f2^-1, f2^2*f3^-1,
          f1^-1*f3*f1*f3^-1, f2^-1*f3*f2*f3^-1, f3^2, f1^-1*f4*f1*f4^-1,
          f2^-1*f4*f2*f4^-1, f3^-1*f4*f3*f4^-1, f4^2 ] ) )

Construction of extensions

The extensions can be constructed using the additional command Extensions and the hand-coded command FrequencySort.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));;
gap> A := TrivialGModule(G,GF(2));;
gap> L := Extensions(G,A);;
gap> FrequencySort(List(L,IdGroup));
[ [ [ 32, 22 ], 2 ], [ [ 32, 23 ], 1 ], [ [ 32, 25 ], 4 ], [ [ 32, 27 ], 4 ],
  [ [ 32, 28 ], 8 ], [ [ 32, 29 ], 4 ], [ [ 32, 30 ], 4 ], [ [ 32, 31 ], 2 ],
  [ [ 32, 34 ], 1 ], [ [ 32, 35 ], 1 ], [ [ 32, 39 ], 2 ], [ [ 32, 40 ], 4 ],
  [ [ 32, 41 ], 2 ], [ [ 32, 42 ], 8 ], [ [ 32, 43 ], 8 ], [ [ 32, 44 ], 8 ],
  [ [ 32, 46 ], 1 ] ]

Under the action of the various automorphism groups

This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := DirectProduct(DihedralGroup(8),CyclicGroup(2));;
gap> A := TrivialGModule(G,GF(2));;
gap> A1 := AutomorphismGroup(G);;
gap> A2 := GL(1,2);;
gap> D := DirectProduct(A1,A2);;
gap> P := CompatiblePairs(G,A,D);;
gap> M := ExtensionRepresentatives(G,A,P);;
gap> List(M,IdGroup);
[ [ 32, 46 ], [ 32, 25 ], [ 32, 27 ], [ 32, 28 ], [ 32, 28 ], [ 32, 34 ],
  [ 32, 40 ], [ 32, 42 ], [ 32, 43 ], [ 32, 44 ], [ 32, 22 ], [ 32, 29 ],
  [ 32, 31 ], [ 32, 30 ], [ 32, 39 ], [ 32, 23 ], [ 32, 35 ], [ 32, 41 ] ]