# Simple fusion system for dihedral group:D8

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This article describes aparticularfusion system on a group of prime power order, namely dihedral group:D8.

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This article describes a particular fusion system on dihedral group:D8 that turns out to be a simple fusion system: it has no normal fusion subsystems. This is the unique simple fusion system on the group.

## Contents

## Explicit description

Equivalence class under isomorphisms, explicit description of subgroups | Subgroups involved | Order | Index | Number of conjugacy classes of subgroups fused | Total number of subgroups (=1 iff weakly closed subgroup for the fusion system) | Are all group automorphisms of each subgroup included? | Size of automorphism group from the fusion system | Total number of isomorphisms (including automorphisms and others) = (number of automorphisms) |
---|---|---|---|---|---|---|---|---|

trivial subgroup | 1 | 8 | 1 | 1 | Yes | 1 | 1 | |

, |
center of dihedral group:D8, non-normal subgroups of dihedral group:D8 (both conjugacy classes) | 2 | 4 | 3 | 5 | Yes | 1 | 25 |

cyclic maximal subgroup of dihedral group:D8 | 4 | 2 | 1 | 1 | Yes | 2 | 2 | |

Klein four-subgroups of dihedral group:D8 (one of them) | 4 | 2 | 1 | 1 | Yes | 6 | 6 | |

Klein four-subgroups of dihedral group:D8 (one of them) | 4 | 2 | 1 | 1 | Yes | 6 | 6 | |

whole group | 8 | 1 | 1 | 1 | No (see below) | 4 | 4 | |

Total | -- | -- | 8 | 10 | -- | -- | -- |

For the whole group, the *only* automorphisms that are included in the fusion system are the inner automorphisms. This can be traced to the fact that the outer automorphism group of the dihedral group is 2-group, and in particular has no 2'-automorphisms.

## Realization in groups

This simple fusion system is realized for the embedding D8 in PSL(3,2). The group PSL(3,2), which is isomorphic to , is a simple non-abelian group of order 168 that has dihedral group:D8 as a 2-Sylow subgroup.

## Element structure

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