Abstract group $C_3$

• Label: 3.1
• Order: \( 3 \)
• Exponent: \( 3 \)
• Abelian: yes
• Simple: yes
• $\card{\operatorname{Aut}(G)}$: \( 2 \)
• Perm deg.: $3$
• Trans deg.: $3$
• Rank: $1$

Group information

Description:	$C_3$
Order:	\(3\)
Exponent:	\(3\)
Automorphism group:	$C_2$, of order \(2\) (generators)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.

Group statistics

Order	1	3
Elements	1	2	3
Conjugacy classes	1	2	3
Divisions	1	1	2
Autjugacy classes	1	1	2

Dimension	1	2
Irr. complex chars.	3	0	3
Irr. rational chars.	1	1	2

Minimal Presentations

Permutation degree:	$3$
Transitive degree:	$3$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	2
Arbitrary	1	2	2

Constructions

Groups of Lie type:	$\ASigmaL(1,3)$
Presentation:	$\langle a \mid a^{3}=1 \rangle$
Permutation group:	$\langle(1,3,2)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ -1 & -1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{2})$
Transitive group:	3T1				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Primary decomposition:	$C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 2 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3$	$G/\operatorname{soc} \simeq$ $C_1$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3: $C_3$ ($C_1$)

Order 1: $C_1$ ($C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3: $C_3$ ($C_1$)

Order 1: $C_1$ ($C_3$)

Series

Derived series	$C_3$	$\rhd$	$C_1$
Chief series	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_3$

Supergroups

This group is a maximal subgroup of 185 larger groups in the database.

This group is a maximal quotient of 235 larger groups in the database.

Character theory

Complex character table

		1A	3A1	3A-1
	Size	1	1	1
	3 P	1A	3A-1	3A1
	Type
3.1.1a	R	$1$	$1$	$1$
3.1.1b1	C	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
3.1.1b2	C	$1$	${\zeta }_3$	${\zeta }_3^{-1}$

Rational character table

		1A	3A
	Size	1	2
	3 P	1A	3A
3.1.1a		$1$	$1$
3.1.1b		$2$	$-1$