Special linear group:SL(2,Z4)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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 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