# Inner automorphism group of wreath product of Z2 and A5

From Groupprops

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## Definition

This group is defined in the following ways:

- It is the inner automorphism group of the wreath product of Z2 and A5.
- The wreath product of Z2 and A5 has center isomorphic to cyclic group:Z2, and this is a direct factor of the whole group. Our group is the other direct factor.
- It is the external semidirect product of elementary abelian group:E16 by alternating group:A5, where the latter acts on the former by taking its standard representation in characteristic two. See also modular representation theory of alternating group:A5.

## Group properties

Property | Meaning | Satisfied? |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

perfect group | Yes | Smallest order example for perfect not implies semisimple |

centerless group | Yes | |

simple non-abelian group | No | |

almost simple group | No | |

quasisimple group | No | |

almost quasisimple group | No | |

semisimple group | No | |

Fitting-free group | No |

## GAP implementation

### Group ID

This finite group has order 960 and has ID 11358 among the groups of order 960 in GAP's SmallGroup library. For context, there are 11,394 groups of order 960. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(960,11358)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(960,11358);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [960,11358]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description | Functions used |
---|---|

InnerAutomorphismsAutomorphismGroup(AutomorphismGroup(WreathProduct(CyclicGroup(2),AlternatingGroup(5)))) |
InnerAutomorphismsAutomorphismGroup, AutomorphismGroup, WreathProduct, CyclicGroup, AlternatingGroup |

PerfectGroup(960,2) |
PerfectGroup |