# Direct product of D8 and Z2

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*This particular group is a finite group of order:* 16

## Definition

This group is defined as the external direct product of the dihedral group of order eight and the cyclic group of order two.

A presentation for it is:

.

## Elements

### Upto conjugacy

There are ten conjugacy classes:

- The identity element. (1)
- The element . (1)
- The element . (1)
- The element . (1)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)

### Upto automorphism

Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, and are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes .

## Group properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

Abelian group | No | and don't commute | |

Nilpotent group | Yes | Prime power order implies nilpotent | |

Metacyclic group | No | ||

Supersolvable group | Yes | Finite nilpotent implies supersolvable | |

Solvable group | Yes | Nilpotent implies solvable | |

T-group | No | , which is normal, but is not normal | Smallest example for normality is not transitive. |

Monolithic group | No | Center is Klein four-group, any subgroup of order two is minimal normal. | |

One-headed group | No | Distinct maximal subgroups of order eight. | |

SC-group | No | ||

ACIC-group | Yes | Every automorph-conjugate subgroup is characteristic | |

Rational group | Yes | Any two elements that generate the same cyclic group are conjugate | Thus, all characters are integer-valued. |

Rational-representation group | Yes | All representations over characteristic zero are realized over the rationals. | Contrast with quaternion group, that is rational but not rational-representation. |