# Special linear group:SL(2,5)

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## Contents

## Definition

This group is defined in the following equivalent ways:

- As the special linear group: is defined as the special linear group of degree two: matrices of determinant over the field of five elements.
- As the binary icosahedral group or binary dodecahedral group.
- As the binary von Dyck group with parameters .
- As the double cover of alternating group for alternating group:A5. In other words, it is the unique stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is alternating group:A5. Viewed this way, it is denoted .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

### Basic arithmetic functions

### Arithmetic functions of a counting nature

## Group properties

### Important properties

Property | 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 alternating group:A5, which is simple. |

almost simple group | No |

### Other properties

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

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 | |

Z-group | No | The 2-Sylow subgroup is the quaternion group, which is not cyclic. |

A-group | No | The 2-Sylow subgroup is the quaternion group, which is not abelian. |

Schur-trivial group | Yes | |

finite group with periodic cohomology | Yes | the 2-Sylow subgroup is the quaternion group and the other Sylow subgroups are cyclic. |

superperfect group | Yes | It is the universal central extension of the perfect group alternating group:A5. |

## Elements

`Further information: element structure of special linear group:SL(2,5)`

### Summary

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

order of the whole group (total number of elements) | 120 |

conjugacy class sizes | 1,1,12,12,12,12,20,20,30 in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time) maximum: 30, number of conjugacy classes: 9, lcm: 60 |

order statistics | 1 of order 1, 1 of order 2, 20 of order 3, 30 of order 4, 24 of order 5, 20 of order 6, 24 of order 10 maximum: 10, lcm (exponent of the whole group): 60 |

## Subgroups

`Further information: subgroup structure of special linear group:SL(2,5)`

### Quick summary

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

number of subgroups | 76 |

number of conjugacy classes of subgroups | 12 |

number of automorphism classes of subgroups | 12 |

isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers | 2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10 5-Sylow: cyclic group:Z5, Sylow number 6 |

Hall subgroups | Other than the whole group, trivial subgroup, and Sylow subgroups, there is a -Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no -Hall subgroup or -Hall subgroup. |

maximal subgroups | There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5). |

normal subgroups | The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5. |

### Subgroup-defining functions

## Linear representation theory

`Further information: linear representation theory of special linear group:SL(2,5)`

### Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,2,2,3,3,4,4,5,6 maximum: 6, lcm: 60, number: 9, sum of squares: 120, quasirandom degree: 2 |

## GAP implementation

### Group ID

This finite group has order 120 and has ID 5 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(120,5)`

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

`gap> G := SmallGroup(120,5);`

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

`IdGroup(G) = [120,5]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be defined using GAP's SpecialLinearGroup function as:

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

SL(2,5) or equivalently SpecialLinearGroup(2,5) |
SL |

PerfectGroup(120) or equivalently PerfectGroup(120,1) |
PerfectGroup |

SchurCover(AlternatingGroup(5)) |
SchurCover, AlternatingGroup |