Wreath product of Z2 and A4
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Contents
Definition
This group is defined as the external wreath product of cyclic group:Z2 by alternating group:A4, where the permutation action of the latter is taken as the natural action on a set of size four.
More explicitly, it is the external semidirect product with base elementary abelian group:E16 (the direct product of four copies of cyclic group:Z2) by alternating group:A4:
where the latter acts on the former by coordinate permutations given by the natural action of on a set of size four.
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 192 | groups with same order | order of semidirect product is product of orders. In this case, we get where is the order of elementary abelian group:E16 and is the order of alternating group:A4. |
exponent of a group | 12 | groups with same order and exponent of a group | groups with same exponent of a group | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
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 192 and has ID 201 among the groups of order 192 in GAP's SmallGroup library. For context, there are 1,543 groups of order 192. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(192,201)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(192,201);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [192,201]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
Description | Functions used |
---|---|
WreathProduct(CyclicGroup(2),AlternatingGroup(4)) | WreathProduct, CyclicGroup, AlternatingGroup |