Center of general linear group:GL(2,3)

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).
The subgroup is a normal subgroup and the quotient group is isomorphic to symmetric group:S4.
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Definition

$G$ is the general linear group of degree two over field:F3. In other words, it is the group of invertible $2\times 2$ matrices with entries over the field of three elements. The field has elements $0,1,2$ with $2=-1$.

$H$ is the subgroup:

$\{{\begin{pmatrix}1&0\\0&1\\\end{pmatrix}},{\begin{pmatrix}2&0\\0&2\\\end{pmatrix}}\}$

$H$ is isomorphic to cyclic group:Z2. It is the center of $G$ (see center of general linear group is group of scalar matrices over center). The quotient group $G/H$ is $PGL(2,3)$ (the projective general linear group of degree two over field:F3) which is isomorphic to symmetric group:S4.

Arithmetic functions

Function	Value	Explanation
order of the whole group	48	order of $GL(2,q)$ is $(q^{2}-1)(q^{2}-q)$. Here $q=3$.
order of the subgroup	2	As center: it has order $q-1$.
index of the subgroup	24	Follows from Lagrange's theorem. Also, index equals order of $PGL(2,q)$, which is $q^{3}-q$.
size of conjugacy class = index of normalizer	1	center is normal
number of conjugacy classes in automorphism class	1	center is characteristic

Subgroup-defining functions

Subgroup-defining function	What it means in general	Why it takes this value
center	set of elements that commute with every group element	center of general linear group is group of scalar matrices over center -- here the underlying ring is a field, which is commutative, so this just gives that the center is the group of scalar matrices over the multiplicative group of the field.
third derived subgroup	derived subgroup of derived subgroup of derived subgroup	derived subgroup is SL(2,3) in GL(2,3), second derived subgroup is Q8 in GL(2,3)
socle	join of all minimal normal subgroups	It is the unique minimal normal subgroup. The group is a monolithic group.
Frattini subgroup	intersection of all maximal subgroups