Direct product of A6 and Z2

From Groupprops
Jump to: navigation, search
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]


This group is defined as the external direct product of alternating group:A6 and cyclic group:Z2.

Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 720 groups with same order order of direct product is product of orders: the direct factors have orders 360 and 2, so the order is 360 \times 2 = 720
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group exponent of direct product is lcm of exponents: the direct factors have exponents 360 and 2 respectively.

GAP implementation

Group ID

This finite group has order 720 and has ID 766 among the groups of order 720 in GAP's SmallGroup library. For context, there are 840 groups of order 720. 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 G:

gap> G := SmallGroup(720,766);

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

IdGroup(G) = [720,766]

or just do:


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

Other descriptions

Description Functions used
DirectProduct(AlternatingGroup(6),CyclicGroup(2)) DirectProduct, AlternatingGroup, CyclicGroup