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G = C3order 3

Cyclic group

p-group, cyclic, elementary abelian, simple, monomial

Aliases: C3, A3, also denoted Z3, Alt3, rotations of a regular triangle, SmallGroup(3,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C3
C1 — C3
C1 — C3
C1 — C3
C1 — C3

Generators and relations for C3
 G = < a | a3=1 >


Character table of C3

 class 13A3B
 size 111
ρ1111    trivial
ρ21ζ3ζ32    linear of order 3 faithful
ρ31ζ32ζ3    linear of order 3 faithful

Permutation representations of C3
Regular action on 3 points - transitive group 3T1
Generators in S3
(1 2 3)

G:=sub<Sym(3)| (1,2,3)>;

G:=Group( (1,2,3) );

G=PermutationGroup([(1,2,3)])

G:=TransitiveGroup(3,1);

Polynomial with Galois group C3 over ℚ
actionf(x)Disc(f)
3T1x3-3x+134

Matrix representation of C3 in GL1(𝔽7) generated by

4
G:=sub<GL(1,GF(7))| [4] >;

C3 in GAP, Magma, Sage, TeX

C_3
% in TeX

G:=Group("C3");
// GroupNames label

G:=SmallGroup(3,1);
// by ID

G=gap.SmallGroup(3,1);
# by ID

G:=PCGroup([1,-3]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^3=1>;
// generators/relations

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