Cyclic group:Z12

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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, denoted $C_{12}, \Z_{12}$ or $\Z/12\Z$, is defined in the following equivalent ways:

1. It is a cyclic group of order $12$.
2. It is the direct product of the cyclic group of order three and the cyclic group of order four.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	12	groups with same order
exponent of a group	12	groups with same order and exponent of a group | groups with same exponent of a group
nilpotency class	1	groups with same order and nilpotency class | groups with same nilpotency class	cyclic implies abelian
derived length	1	groups with same order and derived length | groups with same derived length	cyclic implies abelian
Frattini length	2	groups with same order and Frattini length | groups with same Frattini length
Fitting length	1	groups with same order and Fitting length | groups with same Fitting length

GAP implementation

Group ID

This finite group has order 12 and has ID 2 among the groups of order 12 in GAP's SmallGroup library. For context, there are 5 groups of order 12. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(12,2)

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

gap> G := SmallGroup(12,2);

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

IdGroup(G) = [12,2]

or just do:

IdGroup(G)

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

Other descriptions

Description	Functions used
CyclicGroup(12)	CyclicGroup
DirectProduct(CyclicGroup(4),CyclicGroup(3))	CyclicGroup, DirectProduct