Cyclic group:Z64

Definition

This group is defined as the cyclic group of order $64=2^{6}$.

Arithmetic functions

Function	Value	Explanation
order	64
exponent	64
nilpotency class	1
derived length	1
Fitting length	1
Frattini length	6
minimum size of generating set	1
subgroup rank	1
rank as p-group	1
normal rank as p-group	1
characteristic rank as p-group	1

Group properties

Function	Satisfied?	Explanation
cyclic group	Yes
abelian group	Yes
elementary abelian group	No
metacyclic group	Yes
homocyclic group	Yes
nilpotent group	Yes
group of prime power order	Yes
solvable group	Yes

Subgroups

Since the group is cyclic, there is a unique subgroup of every order dividing its order, which is a cyclic group and the quotient group is cyclic as well. Further information: cyclicity is subgroup-closed, cyclicity is quotient-closed

1. The trivial subgroup. (1)
2. The unique subgroup of order two. Isomorphic to cyclic group:Z2. The quotient group is cyclic group:Z32. (1)
3. The unique subgroup of order four. Isomorphic to cyclic group:Z4. The quotient group is cyclic group:Z16. (1)
4. The unique subgroup of order eight. Isomorphic to cyclic group:Z8. The quotient group is cyclic group:Z8. (1)
5. The unique subgroup of order sixteen. Isomorphic to cyclic group:Z16. The quotient group is cyclic group:Z4. (1)
6. The unique subgroup of order thirty-two. Isomorphic to cyclic group:Z32. The quotient group is cyclic group:Z2. (1)
7. The whole group. (1)

GAP implementation

Group ID

This finite group has order 64 and has ID 1 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,1)

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

gap> G := SmallGroup(64,1);

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

IdGroup(G) = [64,1]

or just do:

IdGroup(G)

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

Other descriptions

The group can be described using GAP's CyclicGroup function:

CyclicGroup(64)