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, or as a semidirect product of the elementary abelian group:E16 and $Z_{2}$. It is given by the following presentation:

$\langle a,b,c,d,x\mid a^{2}=b^{2}=c^{2}=d^{2}=x^{2}=e,ab=ba,ac=ca,ad=da,bc=cb,bd=db,cd=dc,xax^{-1}=ab,xb=bx,xcx^{-1}=cd,xd=dx\rangle$

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

Get more information about groups of the same order at Groups of order 32#The list

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	$32\Gamma _{4}a_{1}$

Arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	32	groups with same order
prime-base logarithm of order	5	groups with same prime-base logarithm of order
exponent of a group	4	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length	2	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	3	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	4	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group	4	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group	4	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group	4	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

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.