SmallGroup(32,9)

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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
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Definition

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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 (32,9), i.e., 9^{th} among groups of order 32
Hall-Senior number 27 among groups of order 32
Hall-Senior symbol 32\Gamma_3c_1

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 2 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 9 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(32,9)

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

gap> G := SmallGroup(32,9);

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

IdGroup(G) = [32,9]

or just do:

IdGroup(G)

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