# Cyclic group:Z12

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## 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