Groups of Order 16

Group of Order 16

Cyclic

$C_{16}$
Alternate Descriptions: (* Most common)

Name Symbol(s)

* Cyclic * C16, Z16

Abelian [16]

GAP ID: [16,1]
Magma ID:?

Presentation: a | a^{16}
Permutation Representations
The following table indicates the degrees of the faithful, transitive, group actions available to the group. This can be thought of as the different ways the group can be embedded in Sn where n is the degree of the group action, and there are no fixed points.

Action Name Degree Generators Stabilizer
(Isomorphism Type) Stabilizer
(Genertors) Primitivity Blocks

Regular 16 (1,...,16) 0 () {1,3,5,7,...},{2,4,6,8,...}

Abelian [8,2]

Elementary Abelian -- Abelian [4,4]

Dihedral

$D_{16}$ $D_{16} - generators$
Alternate Descriptions: (* Most common)

Name Symbol(s)

*Dihedral (Algebraic) *D16

Dihedral (Geometeric) D8

Permutation Representations
The following table indicates the degrees of the faithful, transitive, group actions available to the group. This can be thought of as the different ways the group can be embedded in Sn where n is the degree of the group action, and there are no fixed points.

Action Name Degree Generators Stabilizer
(Isomorphism Type) Stabilizer
(Genertors) Primitivity Blocks

Dihedral 4 (1234),(13) C2 (13) {1,3},{2,4}

Regular 8 0 () ? ?

Automorphism Group: D8

Quaternions

Alternate Descriptions: (* Most common)

Name Symbol(s)

* Quaternions * Q8

Extraspecial ("-" type)

Action Name	Degree	Generators	Stabilizer (Isomorphism Type)	Stabilizer (Genertors)	Primitivity Blocks
Regular	16	(1,...,16)	0	()	{1,3,5,7,...},{2,4,6,8,...}

Name	Symbol(s)
*Dihedral (Algebraic)	*D16
Dihedral (Geometeric)	D8