# Special linear group:SL(2,7)

From Groupprops

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## Contents

## Definition

This group is defined as the special linear group of degree two over field:F7, the field of seven elements.

It is denoted or .

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 336 | groups with same order | As : |

exponent of a group | 168 | groups with same order and exponent of a group | groups with same exponent of a group | |

Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | Follows from special linear group is quasisimple and the fact that the center is isomorphic to cyclic group:Z2. |

### Arithmetic function values of a counting nature

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

number of conjugacy classes | 11 | As , odd: . See element structure of special linear group of degree two. |

number of conjugacy classes of subgroups | 19 | |

number of subgroups | 224 |

## Group properties

Function | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group | No | |

perfect group | Yes | See special linear group is perfect. |

quasisimple group | Yes | See special linear group is quasisimple. The group is perfect, and the inner automorphism group is isomorphic to projective special linear group:PSL(2,7), which is simple. |

almost simple group | No | |

one-headed group | Yes | The center of order two is the unique maximal normal subgroup. |

monolithic group | Yes | The center of order two is the unique minimal normal subgroup. |

T-group | Yes |

## GAP implementation

### Group ID

This finite group has order 336 and has ID 114 among the groups of order 336 in GAP's SmallGroup library. For context, there are 228 groups of order 336. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(336,114)`

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

`gap> G := SmallGroup(336,114);`

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

`IdGroup(G) = [336,114]`

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 |
---|---|

SL(2,7) |
SL |