# Difference between revisions of "Subgroup structure of central product of D8 and Z4"

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# The subgroups <math>\langle a \rangle</math>, <math>\langle xy \rangle</math> and <math>\langle axy \rangle</math>. These are [[normal subgroup]]s but are [[automorphic subgroups]]: they are related by outer automorphisms. Isomorphic to [[cyclic group:Z4]]. (3) | # The subgroups <math>\langle a \rangle</math>, <math>\langle xy \rangle</math> and <math>\langle axy \rangle</math>. These are [[normal subgroup]]s but are [[automorphic subgroups]]: they are related by outer automorphisms. Isomorphic to [[cyclic group:Z4]]. (3) | ||

# The subgroup <math>\langle a^2, x \rangle</math>, <math>\langle a^2, ax \rangle</math> and <math>\langle a^2, ay \rangle</math>. These are all normal subgroups but are related by outer automorphisms. Isomorphic to [[Klein four-group]]. (3) | # The subgroup <math>\langle a^2, x \rangle</math>, <math>\langle a^2, ax \rangle</math> and <math>\langle a^2, ay \rangle</math>. These are all normal subgroups but are related by outer automorphisms. Isomorphic to [[Klein four-group]]. (3) | ||

− | # The subgroup <math>\langle a, xy \rangle</math>. This is an [[isomorph-free subgroup]] of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to [[quaternion group]]. ( | + | # The subgroup <math>\langle a, xy \rangle</math>. This is an [[isomorph-free subgroup]] of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to [[quaternion group]]. (1) |

# The subgroups <math>\langle a,y \rangle</math>, <math>\langle x, y \rangle</math> and <math>\langle ax, y \rangle</math>. These are all normal and related by outer automorphisms. Isomorphic to [[direct product of Z4 and Z2]]. (3) | # The subgroups <math>\langle a,y \rangle</math>, <math>\langle x, y \rangle</math> and <math>\langle ax, y \rangle</math>. These are all normal and related by outer automorphisms. Isomorphic to [[direct product of Z4 and Z2]]. (3) | ||

# The subgroups <math>\langle a,x \rangle</math>, <math>\langle xy, ay \rangle</math> and <math>\langle axy, ay \rangle</math>. These are all normal and are related by outer automorphisms. Isomorphic to [[dihedral group:D8]]. (3) | # The subgroups <math>\langle a,x \rangle</math>, <math>\langle xy, ay \rangle</math> and <math>\langle axy, ay \rangle</math>. These are all normal and are related by outer automorphisms. Isomorphic to [[dihedral group:D8]]. (3) | ||

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===Table classifying isomorphism types of subgroups=== | ===Table classifying isomorphism types of subgroups=== | ||

− | {| class=" | + | {| class="sortable" border="1" |

! Group name !! GAP ID !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup | ! Group name !! GAP ID !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup | ||

|- | |- | ||

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| [[Central product of D8 and Z4]] || <math>(16,13)</math> || 1 || 1 || 1 || 1 | | [[Central product of D8 and Z4]] || <math>(16,13)</math> || 1 || 1 || 1 || 1 | ||

|- | |- | ||

− | + | ! Total || -- || 23 || 20 || 17 || 5 | |

|} | |} | ||

## Latest revision as of 14:33, 6 July 2019

This article gives specific information, namely, subgroup structure, about a particular group, namely: central product of D8 and Z4.

View subgroup structure of particular groups | View other specific information about central product of D8 and Z4

The central product of the dihedral group of order eight and cyclic group of order four is a central product of these groups, over a common central subgroup of order two.

It is given by the presentation:

.

Here, is the dihedral group of order eight and is the cyclic group of order four.

Here is a quick description of all the subgroups of this group:

- The trivial subgroup. Isomorphic to trivial group. (1)
- The subgroup . This is the unique normal subgroup of order two, and is contained in the center. Isomorphic to cyclic group:Z2. (1)
- The subgroups , , , . These come in two conjugacy classes of 2-subnormal subgroups, one comprising and and the other comprising and . However, they are all automorphic subgroups. Isomorphic to cyclic group:Z2. (4)
- The subgroups and . These form a single conjugacy class of 2-subnormal subgroups. Isomorphic to cyclic group:Z2. (2)
- The subgroup of order four. This is the center. Isomorphic to cyclic group:Z4. (1)
- The subgroups , and . These are normal subgroups but are automorphic subgroups: they are related by outer automorphisms. Isomorphic to cyclic group:Z4. (3)
- The subgroup , and . These are all normal subgroups but are related by outer automorphisms. Isomorphic to Klein four-group. (3)
- The subgroup . This is an isomorph-free subgroup of order eight, containing the three non-characteristic cyclic subgroups of order four. Isomorphic to quaternion group. (1)
- The subgroups , and . These are all normal and related by outer automorphisms. Isomorphic to direct product of Z4 and Z2. (3)
- The subgroups , and . These are all normal and are related by outer automorphisms. Isomorphic to dihedral group:D8. (3)
- The whole group. Isomorphic to central product of D8 and Z4. (1)

## Tables for quick information

### Table classifying isomorphism types of subgroups

Group name | GAP ID | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | |

Cyclic group:Z2 | 7 | 4 | 1 | 1 | |

Cyclic group:Z4 | 4 | 4 | 4 | 1 | |

Klein four-group | 3 | 3 | 3 | 0 | |

Direct product of Z4 and Z2 | 3 | 3 | 3 | 0 | |

Dihedral group:D8 | 3 | 3 | 3 | 0 | |

Quaternion group | 1 | 1 | 1 | 1 | |

Central product of D8 and Z4 | 1 | 1 | 1 | 1 | |

Total | -- | 23 | 20 | 17 | 5 |

### Table listing number of subgroups by order

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | |

7 | 4 | 1 | 1 | |

7 | 7 | 7 | 1 | |

7 | 7 | 7 | 1 | |

1 | 1 | 1 | 1 | |

Total | 23 | 20 | 17 | 5 |