Nontrivial semidirect product of Z7 and Z8

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Definition

This group is defined as the external semidirect product of cyclic group:Z7 by cyclic group:Z8, which is not isomorphic to the direct product of Z7 and Z8, that is, the cyclic group:Z56.

Arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	56	groups with same order
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 56 and has ID 1 among the groups of order 56 in GAP's SmallGroup library. For context, there are groups of order 56. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(56,1)

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

gap> G := SmallGroup(56,1);

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

IdGroup(G) = [56,1]

or just do:

IdGroup(G)

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