# Direct product of S3 and Z4

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## Definition

This group is defined as the external direct product of the symmetric group of degree 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 24#Arithmetic functions
Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 24 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
derived length 2 groups with same order and derived length | groups with same derived length
Fitting length 2 groups with same order and Fitting length | groups with same Fitting length
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length
minimum size of generating set 2 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 24 and has ID 5 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(24,5)

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

gap> G := SmallGroup(24,5);

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

IdGroup(G) = [24,5]

or just do:

IdGroup(G)

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