# Special linear group:SL(2,Z4)

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

The group can be defined in the following equivalent ways:

- It is the group or , is defined as the special linear group of degree two over the ring of integers modulo 4.
- It is the group , i.e., the special linear group of degree two over the ring .

## 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

### Basic arithmetic functions

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 48 | groups with same order | As , where is a DVR of length over a field of size : Note that we could take or for the calculation. |

exponent of a group | 12 | groups with same order and exponent of a group | groups with same exponent of a group | |

nilpotency class | -- | not a nilpotent group | |

derived length | 3 | groups with same order and derived length | groups with same derived length | |

Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length |

### Arithmetic functions of a counting nature

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

number of conjugacy classes | 10 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | See element structure of special linear group of degree two over a finite discrete valuation ring |

## GAP implementation

### Group ID

This finite group has order 48 and has ID 30 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(48,30)`

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

`gap> G := SmallGroup(48,30);`

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

`IdGroup(G) = [48,30]`

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,ZmodnZ(4)) |
SL, ZmodnZ |