Nontrivial semidirect product of Z3 and Z8
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Contents
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:
where 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 :
gap> G := SmallGroup(24,1);
Conversely, to check whether a given group 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];