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

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 unitriangular matrix group:UT(3,3) on cyclic group:Z3. The elements of this classify the group extensions with cyclic group:Z3 in the center and unitriangular matrix group:UT(3,3) the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about unitriangular matrix group:UT(3,3) |Get more specific information about cyclic group:Z3

Description of the group

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

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.

Elements

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group
WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group $H^2(G;A)$ for the trivial group action $\leftrightarrow$ congruence classes of central extensions with the specified subgroup $A$ and quotient group $G$.
This descends to a correspondence:
Orbits for the group action of $\operatorname{Aut}(G) \times \operatorname{Aut}(A)$ on $H^2(G;A)$ $\leftrightarrow$ pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.

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.

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 81) Stem extension? Base characteristic in whole group? Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 3) Subgroup information on base in whole group
trivial 1 direct product of UT(3,3) and Z3 12 No No 2 2 3
symmetric and nontrivial 8 SmallGroup(81,3) 3 No  ? 2 2 2
nontrivial 8 SmallGroup(81,9) 9 Yes Yes 3 2 2
nontrivial 16 SmallGroup(81,10) 10 Yes Yes 3 2 2
nontrivial 24 wreath product of Z3 and Z3 7 Yes Yes 3 2 2
nontrivial 24 SmallGroup(81,8) 8 Yes Yes 3 2 2
Total (6 rows) 81 -- -- -- -- -- -- -- --

Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

GAP implementation

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 := <pc group of size 27 with 3 generators>, 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 := <linear mapping by matrix, <vector space of dimension 5 over GF(3)> -> ( GF(3)^4 )>,
presentation := rec( group := <free group on the generators [ f1, f2, f3 ]>, 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 ] ]