# 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

This group is a semidirect product $(Z_8 \times Z_2) \rtimes Z_2$, and can be given by the following presentation:

$\langle a,b,c \mid a^8 = b^2 = x^2 = e, ab = ba, xax^{-1} = ab, xbx^{-1} = a^4 b\rangle$

## 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 $(32,9)$, 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

## 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.