# Finite group that is not isoclinic to a group of smaller order

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A finite group is termed a **finite group that is not isoclinic to a group of smaller order** if there is no finite group of smaller order such that the two groups are isoclinic groups.

For every equivalence class of finite groups up to isocliny, there is a unique minimal order and a (not necessarily unique) representative of that order.

## Examples

Here is a list, for various orders, of finite groups that are not isoclinic to groups of smaller orders. Note that any abelian group is isoclinic to the trivial group, so for any abelianness-forcing number, there are no such finite groups. Where such *minimal* groups are isoclinic to each other, we indicate this:

Order | List of finite groups of that order not isoclinic to groups of smaller order | Number of equivalence classes up to isocliny |
---|---|---|

1 | trivial group | 1 |

2 | -- | 0 |

3 | -- | 0 |

4 | -- | 0 |

5 | -- | 0 |

6 | symmetric group:S3 | 1 |

7 | -- | 0 |

8 | dihedral group:D8, quaternion group (both isoclinic to each other) | 1 |

9 | -- | 0 |

10 | dihedral group:D10 | 1 |

11 | -- | 0 |

12 | alternating group:A4, (isoclinic pair: dihedral group:D12, dicyclic group:Dic12) | 2 |