Nontrivial semidirect product of Z3 and Z8

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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 is defined as the external semidirect product of cyclic group:Z3 by cyclic group:Z8, where the generator of the latter acts on the former via the inverse map. Explicitly, it is given by:

$\langle a,x \mid a^3 = x^8 = e, xax^{-1} = a^{-1} \rangle$

where $e$ denotes the identity element.

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	24	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
Frattini length	3	groups with same order and Frattini length | groups with same Frattini length
Fitting length	2	groups with same order and Fitting length | groups with same Fitting 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 1 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(24,1)

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

gap> G := SmallGroup(24,1);

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

IdGroup(G) = [24,1]

or just do:

IdGroup(G)

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

Description by presentation

Here is a description using the presentation given in the definition:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^3,F.2^8,F.2*F.1*F.2^(-1)*F.1];