# Special linear group:SL(2,Z4)

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## Definition

The group can be defined in the following equivalent ways:

1. It is the group $SL(2,\mathbb{Z}_4)$ or $SL(2,\mathbb{Z}/4\mathbb{Z})$, is defined as the special linear group of degree two over the ring of integers modulo 4.
2. It is the group $SL(2,\mathbb{F}_2[t]/(t^2))$, i.e., the special linear group of degree two over the ring $\mathbb{F}_2[t]/(t^2)$.

## 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 $SL(2,R)$, where $R$ is a DVR of length $l = 2$ over a field of size $q = 2$: $q^{3l - 2}(q^2 - 1) = 2^{3(2) - 2}(2^2 - 1) = (16)(3) = 48$
Note that we could take $R = \mathbb{Z}_4$ or $R = \mathbb{F}_2[t]/(t^2)$ 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 $G$:

gap> G := SmallGroup(48,30);

Conversely, to check whether a given group $G$ 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