SmallGroup(32,33)

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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

This group is defined by the following presentation:

G := \langle a,x,y \mid x^2 = a^4 = y^4 = e, xax^{-1}a^{-1} = y^2, ay = ya, xyx^{-1}y^{-1} = a^2y^2 \rangle

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 for function value
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
max-length of a group 5 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 5 chief length equals prime-base logarithm of order for group of prime power order
composition length 5 composition length equals prime-base logarithm of order for group of prime power 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
Fitting length 1 The group is a nilpotent group, hence its Fitting length is 1. Note that prime power order implies nilpotent, so all groups of the same order have Fitting length 1.
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 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 3 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

Subgroup-defining functions

Subgroup-defining function Isomorphism class Corresponding quotient-defining function Isomorphism class of quotient
center Klein four-group inner automorphism group elementary abelian group:E8
derived subgroup Klein four-group abelianization elementary abelian group:E8
Frattini subgroup Klein four-group Frattini quotient elementary abelian group:E8
socle Klein four-group  ? elementary abelian group:E8
join of abelian subgroups of maximum order direct product of Z4 and Z4  ? cyclic group:Z2
join of abelian subgroups of maximum rank elementary abelian group:E8  ? Klein four-group
ZJ-subgroup direct product of Z4 and Z4  ? cyclic group:Z2

GAP implementation

Group ID

This finite group has order 32 and has ID 33 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(32,33)

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

gap> G := SmallGroup(32,33);

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

IdGroup(G) = [32,33]

or just do:

IdGroup(G)

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