# General linear group:GL(2,3)

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## Contents

## Definition

The general linear group is defined in the following equivalent ways:

- It is the general linear group of degree two: invertible matrices over the field of three elements.
- It is the Schur covering group of symmetric group:S4 of "+" type. The corresponding Schur multiplier (the second homology group is cyclic group:Z2.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 48#Arithmetic functions

### Arithmetic functions of a counting nature

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

number of subgroups | 55 | |

number of conjugacy classes | 8 | |

number of conjugacy classes of subgroups | 16 |

## Group properties

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

Abelian group | No | |

Nilpotent group | No | |

Metacyclic group | No | |

Supersolvable group | No | |

Solvable group | Yes | Length four. |

T-group | No | |

HN-group | No | |

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

One-headed group | Yes | The special linear group. |

## Subgroups

`Further information: Subgroup structure of general linear group:GL(2,3)`

- The trivial group. (1)
- The center, which is a two-element subgroup isomorphic to a cyclic group of order two. It comprises the identity element and its negative. (1)
- The conjugates to the two-element subgroup generated by . (12)
- Subgroups of order three, isomorphic to the cyclic group of order three, all conjugate to the subgroup . (4)
- Subgroups of order four, isomorphic to the cyclic group of order four all conjugate to the subgroup . (3)
- Subgroups of order four, isomorphic to Klein four-group, all conjugate to the subgroup of diagonal matrices. (6)
- Subgroups of order six, isomorphic to the cyclic group of order six, all conjugate to the subgroup . (4)
- Subgroups of order six, isomorphic to the symmetric group of degree three. These are all automorphic to each other, but they come in two conjugacy classes of size four each. An example is .(8)
- A subgroup of order eight, isomorphic to the quaternion group. (1)
- Subgroups of order eight, isomorphic to dihedral group:D8. These are all conjugate subgroups. An example is the the orthogonal group , i.e., the subgroup .(3)
- Subgroups of order eight, isomorphic to cyclic group:Z8. These are all conjugate, and an example is .(3)
- Subgroups of order twelve, isomorphic to dihedral group:D12. These are all conjugate to each other. One example is . (4)
- Subgroups of order sixteen, isomorphic to semidihedral group:SD16. These are all conjugate subgroups and are the 2-Sylow subgroups. (3)
- A unique subgroup of order , namely the special linear group:SL(2,3). (1)
- The whole group. (1)

## Linear representation theory

`Further information: Linear representation theory of general linear group:GL(2,3)`

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Isomorphism class | Comment |
---|---|---|---|

Center | (2) | Cyclic group:Z2 | |

Commutator subgroup | (14) | Special linear group:SL(2,3) | Commutator subgroup of general linear group is special linear group |

Second member of derived series | (9) | Quaternion group | |

Third member of derived series | (2) | Cyclic group:Z2 | |

Socle | (2) | Cyclic group:Z2 | |

Frattini subgroup | (2) | Cyclic group:Z2 | The quotient group, isomorphic to symmetric group:S4, is Frattini-free. |

Fitting subgroup | (9) | Quaternion group |

## Quotient-defining functions

Quotient-defining function | Isomorphism class | Comment |
---|---|---|

Inner automorphism group | symmetric group:S4 | |

Abelianization | cyclic group:Z2 | |

Frattini quotient | symmetric group:S4 | |

Fitting quotient | symmetric group:S3 |

## GAP implementation

### Group ID

This finite group has order 48 and has ID 29 among the groups of order 48 in GAP's SmallGroup library. For context, there are 52 groups of order 48. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(48,29)`

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

`gap> G := SmallGroup(48,29);`

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

`IdGroup(G) = [48,29]`

or just do:

`IdGroup(G)`

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