2-Sylow subgroup 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) semidihedral group:SD16 and the group is (up to isomorphism) general linear group:GL(2,3) (see subgroup structure of general linear group:GL(2,3)).
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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 one of the 2-Sylow subgroups of $G$, i.e.:

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$H$ is isomorphic to semidihedral group:SD16.

Arithmetic functions

Function	Value	Explanation
order of the whole group	48
order of the subgroup	16
index of the subgroup	3
size of conjugacy class of subgroups	3
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 subgroup itself	current page	semidihedral group:SD16
centralizer	$\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \\\end{pmatrix}$	center of general linear group:GL(2,3)	cyclic group:Z2
normal core	$\langle \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 2 & 1 \\\end{pmatrix}$	2-core of general linear group:GL(2,3)	quaternion group
normal closure	the whole group	--	general linear group:GL(2,3)
characteristic core	$\langle \begin{pmatrix} 0 & 1 \\ 2 & 0 \\\end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 2 & 1 \\\end{pmatrix}$	2-core of general linear group:GL(2,3)	quaternion group
characteristic closure	the whole group	--	general linear group:GL(2,3)
commutator with whole group	all matrices of determinant 1	SL(2,3) in GL(2,3)	special linear group:SL(2,3)

Conjugacy class-defining functions

Conjugacy class-defining function	Meaning in general	Why it takes this value
2-Sylow subgroups	$p$-Sylow subgroups are subgroups whose order is a power of $p$ and index is relatively prime to $p$. Sylow subgroups exist and Sylow implies order-conjugate, i.e., any two $p$-Sylow subgroups are conjugate. Here $p = 2$.	The subgroups have order $16 = 2^4$ and index 3.
2-Sylow normalizers	Normalizer of a 2-Sylow subgroup	The 2-Sylow subgroups are self-normalizing subgroups.

Subgroup properties

Sylow and corollaries

The subgroup is a 3-Sylow subgroup, so many properties follow as a corollary of that.

Property	Meaning	Why being Sylow implies the property
order-conjugate subgroup	conjugate to any subgroup of the same order	Sylow implies order-conjugate
isomorph-conjugate subgroup	conjugate to any subgroup isomorphic to it	(via order-conjugate)
automorph-conjugate subgroup	conjugate to any subgroup automorphic to it	(via isomorph-conjugate)
intermediately isomorph-conjugate subgroup	conjugate to any subgroup isomorphic to it inside any intermediate subgroup	Sylow implies intermediately isomorph-conjugate
intermediately automorph-conjugate subgroup	automorph-conjugate in every intermediate subgroup	(via intermediately isomorph-conjugate)
pronormal subgroup	conjugate to any conjugate subgroup in their join	Sylow implies pronormal
weakly pronormal subgroup		(via pronormal)
paranormal subgroup		(via pronormal)
polynormal subgroup		(via pronormal)

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	(see above for other conjugate subgroups)
2-subnormal subgroup	normal subgroup in its normal closure	No
subnormal subgroup	series from subgroup to whole group, each normal in next	No
contranormal subgroup	normal closure is whole group	Yes
self-normalizing subgroup	equals its normalizer in the whole group	Yes
abnormal subgroup		Yes	Sylow normalizer implies abnormal
weakly abnormal subgroup		Yes	(via abnormal)
self-centralizing subgroup	contains its centralizer in the whole group	Yes