# D8 in A6

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) dihedral group:D8 and the group is (up to isomorphism) alternating group:A6 (see subgroup structure of alternating group:A6).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

This article is about the subgroup in the group , where is alternating group:A6, i.e., the alternating group on the set , and is the subgroup:

is a 2-Sylow subgroup of and is isomorphic to dihedral group:D8. There is a total of 45 conjugate subgroups to in (including itself).

## Contents

## Arithmetic functions

Function | Value | Explanation | Comment |
---|---|---|---|

order of the group | 360 | ||

order of the subgroup | 8 | ||

index of the subgroup | 45 | ||

size of conjugacy class | 45 | ||

number of conjugacy classes in automorphism class | 1 |

## Subgroup properties

### Resemblance-based properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

order-conjugate subgroup | conjugate to any subgroup of the same order | Yes | Sylow implies order-conjugate | |

order-dominating subgroup | any subgroup of the whole group whose order divides the order of is contained in a conjugate of | Yes | Sylow implies order-dominating | |

order-dominated subgroup | any subgroup of the whole group whose order is a multiple of the order of contains a conjugate of | Yes | Sylow implies order-dominated | |

order-isomorphic subgroup | isomorphic to any subgroup of the group of the same order | Yes | (via order-conjugate) | |

isomorph-automorphic subgroup | Yes | (via order-conjugate) | ||

automorph-conjugate subgroup | Yes | (via order-conjugate) | ||

order-automorphic subgroup | Yes | (via order-conjugate) | ||

isomorph-conjugate subgroup | Yes | (via order-conjugate) |

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

normal subgroup | equals all its conjugate subgroups | No | (see other conjugate subgroups) | |

subnormal subgroup | No | |||

self-normalizing subgroup | equals its normalizer in the whole group | Yes | ||

abnormal subgroup | Yes | |||

weakly abnormal subgroup | Yes | |||

contranormal subgroup | Yes | |||

maximal subgroup | Yes | |||

pronormal subgroup | Yes | Sylow implies pronormal | ||

weakly pronormal subgroup | Yes | (via pronormal) |

## Fusion system

The subgroup embedding induces the simple fusion system for dihedral group:D8.

## GAP implementation

### Construction of subgroup given group as a black box

Suppose we are already given a group that we know to be isomorphic to alternating group:A6. Then, the subgroup can be constructed using SylowSubgroup as follows:

`H := SylowSubgroup(G,2);`

### Construction of group-subgroup pair

The group and subgroup pair can be defined using GAP's AlternatingGroup and SylowSubgroup functions as follows:

`G := AlternatingGroup(6); H := SylowSubgroup(G,2);`