SmallGroup(32,7)

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is a semidirect product $(Z_{8}\rtimes Z_{2})\rtimes Z_{2}$, and can be given by the presentation:

$\langle a,x,y\mid a^{8}=x^{2}=y^{2}=e,xy=yx,xax^{-1}=a^{5},yay^{-1}=ax\rangle$

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	$(32,7)$, i.e., $7^{th}$ among groups of order 32
Hall-Senior number	47 among groups of order 32
Hall-Senior symbol	$32\Gamma _{7}a_{2}$

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#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	8	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	3	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	3	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	3	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	2	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	3	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	3	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	3	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	2	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	Comment
group of prime power order	Yes
nilpotent group	Yes	prime power order implies nilpotent
supersolvable group	Yes	via nilpotent: finite nilpotent implies supersolvable
solvable group	Yes	via nilpotent: nilpotent implies solvable
abelian group	No

GAP implementation

Group ID

This finite group has order 32 and has ID 7 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(32,7)

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

gap> G := SmallGroup(32,7);

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

IdGroup(G) = [32,7]

or just do:

IdGroup(G)

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