# Special linear group:SL(2,9)

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## Definition

This group is defined in the following equivalent ways:

1. It is the special linear group of degree two over the field of nine elements.
2. It is the group $2 \cdot A_6$, or equivalently, a double cover of alternating group:A6. In other words, it is a stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is alternating group:A6. Because $A_6$ is a perfect group, it is the unique stem extension of this sort. Thus, it belongs to the family double cover of alternating group.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 720#Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 720 groups with same order As $SL(2,q)$, $q = 9$: $q^3 - q = 9^3 - 9 = 720$ (ref: order formulas for linear groups of degree two)
As $2 \cdot A_n, n= 6$: $n! = 6! = 720$
exponent of a group 120 groups with same order and exponent of a group | groups with same exponent of a group As $SL(2,q), q = 9$, underlying prime $p = 3$: $p(q^2 - 1)/2 = 3(9^2 - 1)/2 = 3(80)/2 = 240/2 = 120$ (see element structure of special linear group of degree two over a finite field)
nilpotency class -- not a nilpotent group (true for all $SL(2,q)$)
derived length -- not a solvable group (true for all $SL(2,q)$ except with $q = 2,3$)
Fitting length -- not a solvable group (true for all $SL(2,q)$ except with $q = 2,3$)
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length Frattini subgroup is center, which has order 2 (true for all $SL(2,q)$ with $q$ odd)
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set As $SL(2,q)$, finite $q$: 2
As $2 \cdot A_n$, finite $n$: 2

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation
number of conjugacy classes 13 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $SL(2,q), q = 9$ (odd): $q + 4 = 9 + 4 = 13$; see element structure of special linear group of degree two over a finite field, element structure of special linear group:SL(2,9)#Interpretation as special linear group of degree two

As $2 \cdot A_n, n = 6$: (number of unordered integer partitions of $n$) + 3(number of partitions of $n$ into distinct odd parts) - (number of partitions of $n$ with a positive even number of even parts and with at least one repeated part) $= 11 + 3(1) - 1 = 13$; see element structure of double cover of alternating group, element structure of special linear group:SL(2,9)#Interpretation as double cover of alternating group
number of equivalence classes under rational conjugacy 10 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy
number of subgroups 588 groups with same order and number of subgroups | groups with same number of subgroups See subgroup structure of special linear group:SL(2,9)
number of conjugacy classes of subgroups 27 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups See subgroup structure of special linear group:SL(2,9)

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group No nontrivial center of order two
almost simple group No
quasisimple group Yes
almost quasisimple group Yes
perfect group Yes

## Elements

Further information: element structure of special linear group:SL(2,9)

### Summary

Item Value
order of the whole group (total number of elements) 720
conjugacy class sizes 1,1,40,40,40,40,72,72,72,72,90,90,90
in grouped form: 1 (2 times), 40 (4 times), 72 (4 times), 90 (3 times)
maximum: 90, number of conjugacy classes: 13, lcm: 360
order statistics 1 of order 1, 1 of order 2, 80 of order 3, 90 of order 4, 144 of order 5, 80 of order 6, 180 of order 8, 144 of order 10
maximum: 10, lcm (exponent of the whole group): 120

## Subgroups

Further information: subgroup structure of special linear group:SL(2,9)

### Quick summary

Item Value
number of subgroups 588
number of conjugacy classes of subgroups 27
number of automorphism classes of subgroups PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

## Linear representation theory

Further information: linear representation theory of special linear group:SL(2,9)

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,4,4,5,5,8,8,8,8,9,10,10,10
number: 13, maximum: 10, lcm: 360, sum of squares: 720

## GAP implementation

### Group ID

This finite group has order 720 and has ID 409 among the groups of order 720 in GAP's SmallGroup library. For context, there are 840 groups of order 720. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(720,409)

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

gap> G := SmallGroup(720,409);

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

IdGroup(G) = [720,409]

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(2,9) SL
PerfectGroup(720) or equivalently PerfectGroup(720,1) PerfectGroup
SchurCover(AlternatingGroup(6)) SchurCover, AlternatingGroup