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G = C8order 8 = 23

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C8, also denoted Z8, SmallGroup(8,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C8
C1C2C4 — C8
C1 — C8
C1 — C8
C1C2C2C4 — C8

Generators and relations for C8
 G = < a | a8=1 >


Character table of C8

 class 124A4B8A8B8C8D
 size 11111111
ρ111111111    trivial
ρ21111-1-1-1-1    linear of order 2
ρ31-1i-iζ87ζ85ζ83ζ8    linear of order 8 faithful
ρ411-1-1-ii-ii    linear of order 4
ρ51-1-iiζ85ζ87ζ8ζ83    linear of order 8 faithful
ρ61-1i-iζ83ζ8ζ87ζ85    linear of order 8 faithful
ρ711-1-1i-ii-i    linear of order 4
ρ81-1-iiζ8ζ83ζ85ζ87    linear of order 8 faithful

Permutation representations of C8
Regular action on 8 points - transitive group 8T1
Generators in S8
(1 2 3 4 5 6 7 8)

G:=sub<Sym(8)| (1,2,3,4,5,6,7,8)>;

G:=Group( (1,2,3,4,5,6,7,8) );

G=PermutationGroup([(1,2,3,4,5,6,7,8)])

G:=TransitiveGroup(8,1);

Polynomial with Galois group C8 over ℚ
actionf(x)Disc(f)
8T1x8-8x6+20x4-16x2+2231

Matrix representation of C8 in GL1(𝔽17) generated by

2
G:=sub<GL(1,GF(17))| [2] >;

C8 in GAP, Magma, Sage, TeX

C_8
% in TeX

G:=Group("C8");
// GroupNames label

G:=SmallGroup(8,1);
// by ID

G=gap.SmallGroup(8,1);
# by ID

G:=PCGroup([3,-2,-2,-2,6,16]);
// Polycyclic

G:=Group<a|a^8=1>;
// generators/relations

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