SmallGroup(36,3)

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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 the semidirect product $(Z_{2}\times Z_{2})\rtimes Z_{9}$. Explicitly, it is given by:

$\langle a,b,x\mid a^{2}=b^{2}=x^{9}=e,ab=ba,xax^{-1}=b,xbx^{-1}=ab\rangle$

where $e$ denotes the identity element.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 36#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	36	groups with same order
exponent of a group	18	groups with same order and exponent of a group | groups with same exponent of a group
minimum size of generating set	3	groups with same order and minimum size of generating set | groups with same minimum size of generating set

GAP implementation

Group ID

This finite group has order 36 and has ID 1 among the groups of order 36 in GAP's SmallGroup library. For context, there are groups of order 36. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(36,1)

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

gap> G := SmallGroup(36,1);

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

IdGroup(G) = [36,1]

or just do:

IdGroup(G)

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