# Direct product of Z6 and Z2

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## Definition

This group is defined in the following equivalent ways:

1. It is the direct product of the cyclic group of order six and cyclic group of order two.
2. It is the direct product of the Klein four-group and the cyclic group of order three.
3. It is the direct product of the cyclic group of order three and two copies of the cyclic group of order two.

## 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 6 groups with same order and exponent of a group | groups with same exponent of a group exponent of direct product is lcm of exponents
nilpotency class 1 groups with same order and nilpotency class | groups with same nilpotency class abelian group
derived length 1 groups with same order and derived length | groups with same derived length abelian group
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length direct product of elementary abelian groups
Fitting length 1 groups with same order and Fitting length | groups with same Fitting length abelian group
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group

## GAP implementation

### Group ID

This finite group has order 12 and has ID 5 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(12,5)

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

gap> G := SmallGroup(12,5);

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

IdGroup(G) = [12,5]

or just do:

IdGroup(G)

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