Second cohomology group for trivial group action of SD16 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 semidihedral group:SD16 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and semidihedral group:SD16 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 Klein four-group.
Get more specific information about semidihedral group:SD16 |Get more specific information about cyclic group:Z2|View other constructions whose value is Klein four-group

GAP implementation

Construction of the cohomology group

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

gap> G := SmallGroup(16,8);;
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 := [ [ [ 4, 1 ] ], [ [ 2, 1, 3, 1 ], 0 ],
          [ [ 3, 1, 4, 1 ], [ 3, 1, 4, 1 ], [ 4, 1 ] ],
          [ [ 4, 1 ], [ 4, 1 ], [ 4, 1 ], 0 ] ], orders := [ 2, 2, 2, 2 ],
      wstack := [ [ 4, 1 ], [ 4, 1 ], [ 4, 1 ] ], estack := [  ],
      pstack := [ 3, 3, 3 ], cstack := [ 1, 1, 1 ], mstack := [ 0, 0, 0 ],
      list := [ 1, 0, 0, 1 ],
      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 4 over GF(
    2)> -> ( GF(2)^2 )>,
  presentation := rec( group := <free group on the generators
        [ f1, f2, f3, f4 ]>,
      relators := [ f1^2*f4^-1, f1^-1*f2*f1*f3^-1*f2^-1, f2^2,
          f1^-1*f3*f1*f4^-1*f3^-1, f2^-1*f3*f2*f4^-1*f3^-1, f3^2*f4^-1,
          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.

gap> G := SmallGroup(16,8);;
gap> A := TrivialGModule(G,GF(2));;
gap> L := Extensions(G,A);;
gap> List(L,IdGroup);
[ [ 32, 40 ], [ 32, 9 ], [ 32, 10 ], [ 32, 13 ] ]

Under the action of the various automorphism groups

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

gap> G := SmallGroup(16,8);;
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, 40 ], [ 32, 10 ], [ 32, 9 ], [ 32, 13 ] ]