# Element structure of direct product of D8 and Z2

From Groupprops

This article gives specific information, namely, element structure, about a particular group, namely: direct product of D8 and Z2.

View element structure of particular groups | View other specific information about direct product of D8 and Z2

We use the following presentation, where denotes the identity element:

.

## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Conjugacy class | Size of conjugacy class | Order of elements in conjugacy class | Centralizer of first element of class |
---|---|---|---|

1 | 1 | whole group | |

1 | 2 | whole group | |

1 | 2 | whole group | |

1 | 2 | whole group | |

2 | 4 | -- isomorphic to direct product of Z4 and Z2 | |

2 | 4 | -- isomorphic to direct product of Z4 and Z2 | |

2 | 2 | -- isomorphic to elementary abelian group:E8 | |

2 | 2 | -- isomorphic to elementary abelian group:E8 | |

2 | 2 | -- isomorphic to elementary abelian group:E8 | |

2 | 2 | -- isomorphic to elementary abelian group:E8 |

The equivalence classes up to automorphisms are:

Equivalence class under automorphisms | Size of equivalence class | Number of conjugacy classes in it | Size of each conjugacy class |
---|---|---|---|

1 | 1 | 1 | |

1 | 1 | 1 | |

2 | 2 | 1 | |

4 | 2 | 2 | |

8 | 4 | 2 |

## Order and power information

FACTS TO CHECK AGAINST:

ORDER STATISTICS(cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM(cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

### Order statistics

To compare and contrast the order statistics of this group with other groups of the same order, refer element structure of groups of order 16#Order statistics

Number | Elements of order exactly that number | Number of such elements | Number of conjugacy classes of such elements | Number of elements whose order divides that number | Number of conjugacy classes whose element order divides that number |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | |

2 | 11 | 7 | 12 | 8 | |

4 | 4 | 2 | 16 | 10 |

### Power statistics

Number | powers that are not powers for any larger divisor of the group order | Number of such elements | Number of conjugacy classes of such elements | Number of powers | Number of conjugacy classes of powers |
---|---|---|---|---|---|

1 | 14 | 8 | 16 | 10 | |

2 | 1 | 1 | 2 | 2 | |

4 | -- | 0 | 0 | 1 | 1 |

8 | -- | 0 | 0 | 1 | 1 |

16 | 1 | 1 | 1 | 1 |