General linear group:GL(2,3)
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Contents
Definition
The general linear group is defined in the following equivalent ways:
 It is the general linear group of degree two: invertible matrices over the field of three elements.
 It is the Schur covering group of symmetric group:S4 of "+" type. The corresponding Schur multiplier (the second homology group is cyclic group:Z2.
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
Arithmetic functions of a counting nature
Function  Value  Explanation 

number of subgroups  55  
number of conjugacy classes  8  
number of conjugacy classes of subgroups  16 
Group properties
Property  Satisfied  Explanation 

Abelian group  No  
Nilpotent group  No  
Metacyclic group  No  
Supersolvable group  No  
Solvable group  Yes  Length four. 
Tgroup  No  
HNgroup  No  
Monolithic group  Yes  The center of order two is the unique minimal normal subgroup. 
Oneheaded group  Yes  The special linear group. 
Subgroups
Further information: Subgroup structure of general linear group:GL(2,3)
 The trivial group. (1)
 The center, which is a twoelement subgroup isomorphic to a cyclic group of order two. It comprises the identity element and its negative. (1)
 The conjugates to the twoelement subgroup generated by . (12)
 Subgroups of order three, isomorphic to the cyclic group of order three, all conjugate to the subgroup . (4)
 Subgroups of order four, isomorphic to the cyclic group of order four all conjugate to the subgroup . (3)
 Subgroups of order four, isomorphic to Klein fourgroup, all conjugate to the subgroup of diagonal matrices. (6)
 Subgroups of order six, isomorphic to the cyclic group of order six, all conjugate to the subgroup . (4)
 Subgroups of order six, isomorphic to the symmetric group of degree three. These are all automorphic to each other, but they come in two conjugacy classes of size four each. An example is .(8)
 A subgroup of order eight, isomorphic to the quaternion group. (1)
 Subgroups of order eight, isomorphic to dihedral group:D8. These are all conjugate subgroups. An example is the the orthogonal group , i.e., the subgroup .(3)
 Subgroups of order eight, isomorphic to cyclic group:Z8. These are all conjugate, and an example is .(3)
 Subgroups of order twelve, isomorphic to dihedral group:D12. These are all conjugate to each other. One example is . (4)
 Subgroups of order sixteen, isomorphic to semidihedral group:SD16. These are all conjugate subgroups and are the 2Sylow subgroups. (3)
 A unique subgroup of order , namely the special linear group:SL(2,3). (1)
 The whole group. (1)
Linear representation theory
Further information: Linear representation theory of general linear group:GL(2,3)
Subgroupdefining functions
Subgroupdefining function  Subgroup type in list  Isomorphism class  Comment 

Center  (2)  Cyclic group:Z2  
Commutator subgroup  (14)  Special linear group:SL(2,3)  "speciallineargroup:SL(" can not be assigned to a declared number type with value 2. Commutator subgroup of general linear group is special linear group

Second member of derived series  (9)  Quaternion group  
Third member of derived series  (2)  Cyclic group:Z2  
Socle  (2)  "cyclicgroup:Z" can not be assigned to a declared number type with value 2. Cyclic group:Z2 

Frattini subgroup  (2)  "cyclicgroup:Z" can not be assigned to a declared number type with value 2. Cyclic group:Z2 
The quotient group, isomorphic to symmetric group:S4, is Frattinifree. 
Fitting subgroup  (9)  "quaternion group" is not a number. Quaternion group 
Quotientdefining functions
Quotientdefining function  Isomorphism class  Comment 

Inner automorphism group  symmetric group:S4  
Abelianization  cyclic group:Z2  
Frattini quotient  symmetric group:S4  
Fitting quotient  symmetric group:S3 
GAP implementation
Group ID
This finite group has order 48 and has ID 29 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(48,29)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(48,29);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [48,29]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.