Direct product of A5 and Z2
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Contents
Definition
This group is defined in the following equivalent ways:
- It is the full icosahedral group: it is the group of all rigid symetries of the regular icosahedron, including both orientation-preserving symmetries and orientation-reversing symmetries.
- It is the external direct product of the alternating group of degree five and the cyclic group of order two.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions
Basic arithmetic functions
Function | Value | Similar groups | Explanation for function value |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 120 | groups with same order | order of direct product is product of orders, so the order is |
exponent of a group | 30 | groups with same order and exponent of a group | groups with same exponent of a group | exponent of direct product is lcm of exponents, so lcm of exponents of and , which is |
composition length | 2 | groups with same order and composition length | groups with same composition length | direct product of two simple groups |
chief length | 2 | groups with same order and chief length | groups with same chief length | direct product of two simple groups |
max-length | 5 | groups with same order and max-length | groups with same max-length | |
Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | |
nilpotency class | not a nilpotent group | ||
derived length | not a solvable group | ||
Fitting length | not a solvable group |
Group properties
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No | |
nilpotent group | No | |
solvable group | No | |
simple group, simple non-abelian group | No | |
almost simple group | No | |
quasisimple group | No | |
almost quasisimple group | No | |
semisimple group | No | |
perfect group | No | |
directly indecomposable group | No |
GAP implementation
Group ID
This finite group has order 120 and has ID 35 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(120,35)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(120,35);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [120,35]
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 |
---|---|
DirectProduct(AlternatingGroup(5),CyclicGroup(2)) | DirectProduct, AlternatingGroup, CyclicGroup |