# SmallGroup(32,27)

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## Definition

This group is a semidirect product of elementary abelian group:E8 and Klein four-group where the latter acts faithfully by transvections relative to a particular plane. It is given by the following presentation:

$\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle$

It can also be described as the subgroup of upper-triangular unipotent matrix group:U(4,2) given by matrices with the $(1,2)$-entry equal to zero, i.e., matrices of the form:

$\begin{pmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$

It can also be defined as the 2-Sylow subgroup of the automorphism group of the homocyclic group given as the direct product of Z4 and Z4.

Another group that occurs as a faithful semidirect product of the elementary abelian group of order eight and the Klein four-group is SmallGroup(32,49).

## Position in classifications

Type of classification Position/number in classification
GAP ID $(32,27)$, i.e., $27^{th}$ among groups of order 32
Hall-Senior number 33 among groups of order 32
Hall-Senior symbol $32\Gamma_4a_1$

## Group properties

Property Satisfied? Explanation
Cyclic group No
Abelian group No
Metacyclic group No
Metabelian group Yes Has elementary abelian maximal subgroup
Group of nilpotency class two Yes Derived subgroup is the plane of translation, which is in the center

## GAP implementation

### Group ID

This finite group has order 32 and has ID 27 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,27)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(32,27);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [32,27]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.