# General linear group:GL(2,3)

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## Definition

The general linear group $GL(2,3)$ is defined in the following equivalent ways:

1. It is the general linear group of degree two: $2 \times 2$ invertible matrices over the field of three elements.
2. It is the Schur covering group of symmetric group:S4 of "+" type. The corresponding Schur multiplier (the second homology group $H_2(S_4;\mathbb{Z})$ 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
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 48 groups with same order Order of $GL(2,q)$ is $(q^2 - 1)(q^2 -q)$. Here, $q = 3$, and we get $(3^2 - 1)(3^2 - 3) = 48$.
exponent of a group 24 groups with same order and exponent of a group | groups with same exponent of a group Elements of order $2,3,4,6,8$.
derived length 4 groups with same order and derived length | groups with same derived length derived subgroup is SL(2,3) (See SL(2,3) in GL(2,3), second derived subgroup is isomorphic to quaternion group (see Q8 in GL(2,3)), third derived subgroup is center of general linear group:GL(2,3), which is abelian.
nilpotency class -- -- not a nilpotent group.
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length Intersection of maximal subgroups is center of general linear group:GL(2,3).
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group --
max-length of a group 5 groups with same order and max-length of a group | groups with same max-length of a group
composition length 5 groups with same order and composition length | groups with same composition length
chief length 4 groups with same order and chief length | groups with same chief length

### 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.
T-group No
HN-group No
Monolithic group Yes The center of order two is the unique minimal normal subgroup.
One-headed group Yes The special linear group.

## Subgroups

Further information: Subgroup structure of general linear group:GL(2,3)

1. The trivial group. (1)
2. The center, which is a two-element subgroup isomorphic to a cyclic group of order two. It comprises the identity element and its negative. (1)
3. The conjugates to the two-element subgroup generated by $\begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}$. (12)
4. Subgroups of order three, isomorphic to the cyclic group of order three, all conjugate to the subgroup $\langle \begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}\rangle$. (4)
5. Subgroups of order four, isomorphic to the cyclic group of order four all conjugate to the subgroup $\langle \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}\rangle$. (3)
6. Subgroups of order four, isomorphic to Klein four-group, all conjugate to the subgroup of diagonal matrices. (6)
7. Subgroups of order six, isomorphic to the cyclic group of order six, all conjugate to the subgroup $\langle \begin{pmatrix}2 & 2 \\ 0 & 2 \\\end{pmatrix}\rangle$. (4)
8. 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 $\langle \begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix} \rangle$.(8)
9. A subgroup of order eight, isomorphic to the quaternion group. (1)
10. Subgroups of order eight, isomorphic to dihedral group:D8. These are all conjugate subgroups. An example is the the orthogonal group $O(2,3)$, i.e., the subgroup $\langle \begin{pmatrix}0 & 1 \\ 1 & 0 \\\end{pmatrix}, \begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}\rangle$.(3)
11. Subgroups of order eight, isomorphic to cyclic group:Z8. These are all conjugate, and an example is $\langle \begin{pmatrix} 1 & 1 \\ 2 & 1 \\\end{pmatrix}\rangle$.(3)
12. Subgroups of order twelve, isomorphic to dihedral group:D12. These are all conjugate to each other. One example is $\langle \begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}, \begin{pmatrix}-1 & 0 \\ 0 & 1 \\\end{pmatrix} \rangle$. (4)
13. Subgroups of order sixteen, isomorphic to semidihedral group:SD16. These are all conjugate subgroups and are the 2-Sylow subgroups. (3)
14. A unique subgroup of order $24$, namely the special linear group:SL(2,3). (1)
15. The whole group. (1)

## Linear representation theory

Further information: Linear representation theory of general linear group:GL(2,3)

## Subgroup-defining functions

Subgroup-defining function Subgroup type in list Isomorphism class Comment
Center (2) Cyclic group:Z2
Commutator subgroup (14) Special linear group:SL(2,3) 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) Cyclic group:Z2
Frattini subgroup (2) Cyclic group:Z2 The quotient group, isomorphic to symmetric group:S4, is Frattini-free.
Fitting subgroup (9) Quaternion group

## Quotient-defining functions

Quotient-defining 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 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 $G$:

gap> G := SmallGroup(48,29);

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

IdGroup(G) = [48,29]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.