Special linear group:SL(2,9)
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Contents
Definition
This group is defined in the following equivalent ways:
- It is the special linear group of degree two over the field of nine elements.
- It is the group , 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 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 , : (ref: order formulas for linear groups of degree two) As : |
exponent of a group | 120 | groups with same order and exponent of a group | groups with same exponent of a group | As , underlying prime : (see element structure of special linear group of degree two over a finite field) |
nilpotency class | -- | not a nilpotent group (true for all ) | |
derived length | -- | not a solvable group (true for all except with ) | |
Fitting length | -- | not a solvable group (true for all except with ) | |
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 with 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 , finite : 2 As , finite : 2 |
Arithmetic functions of a counting nature
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 or ) | 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 :
gap> G := SmallGroup(720,409);
Conversely, to check whether a given group 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 |