Wreath product of A4 and Z2
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This group is defined as the external wreath product of alternating group:A4 and cyclic group:Z2, where the latter acts via the regular group action. More explicitly it is the external semidirect product:
where the non-identity element of acts by the coordinate exchange automorphism on .
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 288#Arithmetic functions
|order (number of elements, equivalently, cardinality or size of underlying set)||288||groups with same order||order of semidirect product is product of orders: order is , where is the order of alternating group:A4.|
This finite group has order 288 and has ID 1025 among the groups of order 288 in GAP's SmallGroup library. For context, there are 1,045 groups of order 288. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(288,1025);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [288,1025]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
|WreathProduct(AlternatingGroup(4),CyclicGroup(2))||WreathProduct, AlternatingGroup, CyclicGroup|