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.

Short descriptions

Description Functions used Mathematical comments
DirectProduct(SymmetricGroup(3),CyclicGroup(4)) SymmetricGroup, CyclicGroup