# Schur cover of alternating group:A6

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## Definition

This group, termed the **Schur cover of alternating group:A6** and sometimes denoted , is defined in the following equivalent ways:

- It is the unique quasisimple group with center cyclic group:Z6 and quotient group alternating group:A6.
- It is the Schur covering group of alternating group:A6.
- It is the Schur covering group of special linear group:SL(2,9).

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2160#Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 2160 | groups with same order | As the Schur covering group of : order of Schur multiplier of (which is 6) times order of (which is ) |

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group, simple non-abelian group | No | nontrivial center of order six |

almost simple group | No | |

quasisimple group | Yes | |

almost quasisimple group | Yes | |

perfect group | Yes |

## GAP implementation

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

SchurCover(AlternatingGroup(6)) |
SchurCover, AlternatingGroup |

SchurCover(SL(2,9)) |
SchurCover, SL |

PerfectGroup(2160) or equivalently PerfectGroup(2160,1) |
PerfectGroup |