Cyclic group:Z64

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

As an abelian group of prime power order
This group is the abelian group of prime power order corresponding to the partition:

$$\! 6 = 6$$

In other words, it is the group $$\mathbb{Z}_{p^6}$$.

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.


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

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

CyclicGroup(64)