Nontrivial semidirect product of Z5 and Q8

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 of order 40 is a semidirect product $Z_{5}\rtimes Q_{8}$, defined by means of the following presentation:

$\langle a,i,j\mid a^{5}=i^{4}=j^{4}=e,iji^{-1}=j^{-1},ia=ai,jaj^{-1}=a^{-1},\rangle$

GAP implementation

Group ID

This finite group has order 40 and has ID 4 among the groups of order 40 in GAP's SmallGroup library. For context, there are groups of order 40. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(40,4)

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

gap> G := SmallGroup(40,4);

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

IdGroup(G) = [40,4]

or just do:

IdGroup(G)

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