# dicyclic groupsの例文

- The dicyclic group has a unique center of Dic " n ".
- When " m " = 1 this group is a binary dihedral or dicyclic group.
- Note that the dicyclic group does not contain any subgroup isomorphic to Dih " n ".
- More abstractly, one can define the dicyclic group Dic " n " as any group having the presentation
- For this reason the dicyclic group is also known as the "'binary dihedral group " '.
- Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism.
- More generally, when " n " is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group.
- There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of " mirroring " of an underlying cyclic group.
- The analogous pre-image construction, using Pin + ( 2 ) instead of Pin & minus; ( 2 ), yields another dihedral group, Dih 2 " n ", rather than a dicyclic group.
- In 2 dimensions, the distinction between Pin + and Pin " mirrors the distinction between the dihedral group of a 2 " n "-gon and the dicyclic group of the cyclic group " C " 2 " n ".
- The dicyclic group is a binary polyhedral group it is one of the classes of subgroups of the Pin group Pin & minus; ( 2 ), which is a subgroup of the Spin group Spin ( 3 ) and in this context is known as the "'binary dihedral group " '.
- The smallest abstract groups that are " not " any symmetry group in 3D, are the quaternion group ( of order 8 ), Z 3 & times; Z 3 ( of order 9 ), the dicyclic group Dic 3 ( of order 12 ), and 10 of the 14 groups of order 16.