# Projective special linear group:PSL(2,11)

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## Contents

## Definition

This group is defined as the projective special linear group of degree two over field:F11, the field with 11 elements.

## Arithmetic functions

### Basic arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 660 | groups with same order | As , : |

exponent of a group | 330 | groups with same order and exponent of a group | groups with same exponent of a group | As , underlying prime (odd): |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | the group is a simple non-abelian group |

chief length | 1 | groups with same order and chief length | groups with same chief length | the group is a simple non-abelian group |

composition length | 1 | groups with same order and composition length | groups with same composition length | the group is a simple non-abelian group |

### Arithmetic functions of an element-counting nature

### Arithmetic functions of a subgroup-counting nature

## Group properties

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

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group, simple non-abelian group | Yes | projective special linear group is simple (with a couple of exceptions, but this isn't one of them) |

minimal simple group | No | contains a subgroup isomorphic to alternating group:A5 (really?). See also classification of finite minimal simple groups |

## Subgroups

`Further information: subgroup structure of projective special linear group:PSL(2,11)`

### Quick summary

Item | Value |
---|---|

Number of subgroups | 620 |

Number of conjugacy classes of subgroups | 16 |

Number of automorphism classes of subgroups | 14 |

Isomorphism classes of Sylow subgroups and the corresponding fusion systems | 2-Sylow: Klein four-group, fusion system is simple fusion system for Klein four-group, Sylow number is 55 3-Sylow: cyclic group:Z3, fusion system is the non-inner one, Sylow number is 55 5-Sylow: cyclic group:Z5, Sylow number is 66 11-Sylow: cyclic group:Z11, Sylow number is 12 |

Hall subgroups | Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups (of order 12) and -Hall subgroups (of order 60), and -Hall subgroups (of order 55) Note that there are two distinct isomorphism classes of -Hall subgroups: dihedral group:D12 and alternating group:A4. This gives the smallest example illustrating that Hall not implies order-isomorphic |

maximal subgroups | maximal subgroups have orders 12, 55, 60. |

normal subgroups | only the whole group and the trivial subgroup, because the group is simple. See projective special linear group is simple (with a couple of small exceptions, but this isn't one of them) |

subgroups that are simple non-abelian groups (apart from the whole group itself) | alternating group:A5 (order 60) |

## Linear representation theory

`Further information: linear representation theory of projective special linear group:PSL(2,11)`

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (such as or ) | 1,5,5,10,10,11,12,12 in grouped form: 1 (1 time), 5 (2 times), 10 (2 times), 11 (1 time), 12 (2 times) number: 8, sum of squares: 660, maximum: 12, quasirandom degree: 5, lcm: 660 |

number of irreducible representations (equals number of conjugacy classes) | 8 As ( odd): |

## GAP implementation

### Group ID

This finite group has order 660 and has ID 13 among the groups of order 660 in GAP's SmallGroup library. For context, there are 41 groups of order 660. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(660,13)`

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

`gap> G := SmallGroup(660,13);`

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

`IdGroup(G) = [660,13]`

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 |
---|---|

PSL(2,11) |
PSL |

PerfectGroup(660) or equivalently PerfectGroup(660,1) |
PerfectGroup |