# Supergroups of Klein four-group

This article gives specific information, namely, supergroups, about a particular group, namely: Klein four-group.

View supergroups of particular groups | View other specific information about Klein four-group

This article discusses possible supergroups of the Klein four-group.

Note that unlike the discussion of the subgroup structure of Klein four-group, thisdiscussion is necessarily not comprehensive, because there are infinitely many groups containing the Klein four-group as a subgroup.

## General lists

### Direct products

Each of the groups listed below arises as the external direct product of Klein four-group and some nontrivial group. In particular, each of these groups contains the Klein four-group as a direct factor -- and hence as both a normal subgroup and a quotient group.

Note that since order of direct product is product of orders, if the other group has order , the direct product has order .

Other factor of direct product | Order | Second part of GAP ID | Value of direct product | Order | Second part of GAP ID | Hall-Senior symbol (if applicable) |
---|---|---|---|---|---|---|

cyclic group:Z2 | 2 | 1 | elementary abelian group:E8 | 8 | 5 | |

cyclic group:Z3 | 3 | 1 | direct product of Z6 and Z2 | 12 | 5 | |

cyclic group:Z4 | 4 | 1 | direct product of Z4 and V4 | 16 | 10 | |

Klein four-group | 4 | 2 | elementary abelian group:E16 | 16 | 14 | |

cyclic group:Z5 | 5 | 1 | direct product of Z10 and Z2 | 20 | 5 | |

symmetric group:S3 | 6 | 1 | direct product of D12 and Z2 | 24 | 14 | |

cyclic group:Z6 | 6 | 2 | direct product of E8 and Z3 | 24 | 15 |

## Classification of supergroups of order eight

### The general procedure

Suppose is a group of order eight containing a subgroup of order four isomorphic to the Klein four-group. Note that since is of index two, it is a normal subgroup. Further, the quotient group is isomorphic to , the cyclic group of order two.

The classification has two steps:

- Find all homomorphisms from to .
- For each such homomorphism, find all possible extensions. These are classified by the elements of for the action.

### The classification

is isomorphic to the symmetric group of degree three. This has three subgroups of order two, all of which are conjugate. Thus, up to equivalence under automorphisms of , there is only one subgroup of order two.

Thus, there are (up to equivalence under automorphisms) two homomorphisms from to : the trivial homomorphism and an isomorphism to one of the subgroups of order two.

For the trivial map, the cohomology group has order two, and the two extensions are:

- elementary abelian group of order eight: This corresponds to the identity element of .
- direct product of Z4 and Z2: This corresponds to the non-identity element of .

For the nontrivial map, the cohomology group is trivial, and the unique extension corresponding to it is the dihedral group of order eight.