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:

$\langle x,y,z,a,b\mid x^{2}=y^{2}=z^{2}=a^{2}=b^{2}=e,xy=yx,xz,=zx,yz=zy,aza^{-1}=xz,bzb^{-1}=yz\rangle$

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:

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

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).

Arithmetic functions

Function	Value	Explanation
order	32]]
exponent	4
nilpotency class	2
derived length	2
Frattini length	2
Fitting length	1
minimum size of generating set	3
rank as p-group	4
normal rank	4
characteristic rank	4

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 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.