Second cohomology group for trivial group action of UT(3,3) on Z3

Description of the group
This article describes the second cohomology group for trivial group action:

$$\! H^2(G;A)$$

where $$A$$ is cyclic group:Z3 and $$G \cong UT(3,3)$$ is the group unitriangular matrix group:UT(3,3), defined as the group of upper-triangular unipotent $$3 \times 3$$ matrices over field:F3. It is the only non-abelian group of order 27 and exponent 3.

The group is isomorphic to elementary abelian group:E81.

Related groups

 * Second cohomology group for trivial group action of UT(3,p) on Zp (note that the $$p = 3$$ case, which is the subject of this page, differs somewhat from the $$p \ge 5$$ cases)
 * Second cohomology group for trivial group action of D8 on Z2

Elements
Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z3 and the quotient group isomorphic to prime-cube order group:U(3,3). Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is $$3 \times 27 = 81$$.

The minimum size of generating set of the extension group is at least equal to 2 (which is the minimum size of generating set of the quotient group) and at most equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

The nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

General background
We know from the general theory that there is a natural short exact sequence:

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \to \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0$$

where $$G^{\operatorname{ab}}$$ is the abelianization of $$G$$ and its image comprises those extensions where the restricted extension of the derived subgroup $$[G,G]$$ on $$A$$ is trivial and the corresponding extension of the quotient group is abelian. Also, $$H_2(G;\mathbb{Z}) = M(G)$$ is the Schur multiplier of $$G$$.

We also know, again from the general theory, that the short exact sequence above splits, i.e., the image of $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ has a complement inside $$H^2(G;A)$$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case
For this choice of $$G$$ and $$A$$, $$G^{\operatorname{ab}}$$ is isomorphic to elementary abelian group:E9. The corresponding group $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ is also elementary abelian group:E9.

The Schur multiplier $$H_2(G;\mathbb{Z})$$ is elementary abelian group:E9, hence $$\operatorname{Hom}(H_2(G;\mathbb{Z}),A)$$ is also isomorphic to elementary abelian group:E9.

The image of $$\operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A)$$ in $$H^2(G;A)$$ comprises the nine non-stem extensions (the trivial extensions and the eight extensions giving SmallGroup(81,3)). It has nine cosets in the whole group.

Construction of the cohomology group
The cohomology group can be constructed using the GAP functions TwoCohomology, TrivialGModule, GF:

gap> G := SmallGroup(27,3);; gap> A := TrivialGModule(G,GF(3));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(3), isMTXModule := true, dimension := 1, generators := [ [ [ Z(3)^0 ] ], [ [ Z(3)^0 ] ], [ [ Z(3)^0 ] ] ] ), collector := rec( relators := [ [ 0 ], [ [ 2, 1, 3, 1 ], 0 ], [ [ 3, 1 ], [ 3, 1 ], 0 ] ], orders := [ 3, 3, 3 ], wstack := [ [ 1, 1 ], [ 3, 1 ], [ 2, 1, 3, 1 ] ], estack := [ ], pstack := [ 3, 3, 5 ], cstack := [ 1, 1, 1 ], mstack := [ 0, 0, 0 ], list := [ 0, 0, 0 ], module := [ [ [ Z(3)^0 ] ], [ [ Z(3)^0 ] ], [ [ Z(3)^0 ] ] ], mone := [ [ Z(3)^0 ] ], mzero := [ [ 0*Z(3) ] ], avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6 ] ), cohom :=  -> ( GF(3)^4 )>, presentation := rec( group := , relators := [ f1^3, f1^-1*f2*f1*f3^-1*f2^-1, f2^3, f1^-1*f3*f1*f3^-1, f2^-1*f3*f2*f3^-1, f3^3 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions and the hand-coded function FrequencySort, along with IdGroup:

gap> G := SmallGroup(27,3);; gap> A := TrivialGModule(G,GF(3));; gap> L := Extensions(G,A);; gap> K := List(L,IdGroup);; gap> FrequencySort(K); [ [ [ 81, 3 ], 8 ], [ [ 81, 7 ], 24 ], [ [ 81, 8 ], 24 ], [ [ 81, 9 ], 8 ], [ [ 81, 10 ], 16 ], [ [ 81, 12 ], 1 ] ]