2-core 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) quaternion group 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:S3.
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Definition

$G$ is the general linear group:GL(2,3), i.e., the general linear group of degree two over field:F3. In other words, it is the group of invertible $2 \times 2$ matrices over the field with three elements. The field has elements 0,1,2, with $2 = -1$.

$H$ is the subgroup:

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

$H$ is a normal subgroup of $G$ and is isomorphic to the quaternion group of order 8. The quotient group is isomorphic to symmetric group:S3.

Arithmetic functions

Function	Value	Explanation
order of the whole group	48	Order of $GL(2,q)$ is $(q^2 - q)(q^2 - 1)$. Here $q = 3$.
order of the subgroup	8
index of the subgroup	6
size of conjugacy class = index of normalizer	1
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the whole group	--	general linear group:GL(2,3)
centralizer	$\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}\}$	center of general linear group:SL(2,3)	cyclic group:Z2
normal core	the subgroup itself	current page	quaternion group
normal closure	the subgroup itself	current page	quaternion group
characteristic core	the subgroup itself	current page	quaternion group
characteristic closure	the subgroup itself	current page	quaternion group
commutator with whole group	the subgroup itself	current page	quaternion group

Subgroup-defining functions

The subgroup is a characteristic subgroup of the whole group and arises as the result of many subgroup-defining functions. Some of these are given below.

Subgroup-defining function	Meaning in general	Why it takes this value
second derived subgroup	derived subgroup of derived subgroup	The derived subgroup is special linear group:SL(2,3) (see SL(2,3) in GL(2,3)) and the derived subgroup of that is this subgroup (see Q8 in SL(2,3)).
Fitting subgroup	join of all nilpotent normal subgroups	The only nilpotent normal subgroups of the whole group are this subgroup, the trivial subgroup, and the center of the whole group (which is also the center of the subgroup).
2-Sylow core or 2-core	largest normal subgroup whose order is a power of 2; normal core of any 2-Sylow subgroup	There are three 2-Sylow subgroups, each of order 16 and isomorphic to semidihedral group:SD16 (See SD16 in GL(2,3)). Their intersection is precisely this subgroup.