Cyclic Group of order <i>p</i> White Sheet

Cyclic group of order p White Sheet

[Class] | [Family] | Samples by Order: [8], [27].

Let p be any prime.

Subgroup Lattice

$C_{p^3}$

Alternate Descriptions: (* Most common)

Name Symbol(s)

* Cyclic * Cp^3, Zp^3

Abelian [p^3]

GAP ID:?
Magma ID:?

Standard Invariants:

Invariant Value

Abelian True

Nilpotent True

Nilpotency Class 1

Ascending Central Series [0,Cp^3]

Descending Central Series [Cp^3,0]

Solvable True

Composition Factors {Cp, Cp, Cp}

Automorphism Group p=2
C2 x C2
p>2
Cp-1 x Cp^2

Invariant	Value
Abelian	True
Nilpotent	True
Nilpotency Class	1
Ascending Central Series	[0,Cp^3]
Descending Central Series	[Cp^3,0]
Solvable	True
Composition Factors	{Cp, Cp, Cp}
Automorphism Group	p=2 C2 x C2 p>2 Cp-1 x Cp^2