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G = C7order 7

Cyclic group

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

Aliases: C7, also denoted Z7, SmallGroup(7,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C7
C1 — C7
C1 — C7
C1 — C7
C1 — C7

Generators and relations for C7
 G = < a | a7=1 >


Character table of C7

 class 17A7B7C7D7E7F
 size 1111111
ρ11111111    trivial
ρ21ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7 faithful
ρ31ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7 faithful
ρ41ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7 faithful
ρ51ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7 faithful
ρ61ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7 faithful
ρ71ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7 faithful

Permutation representations of C7
Regular action on 7 points - transitive group 7T1
Generators in S7
(1 2 3 4 5 6 7)

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

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

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

G:=TransitiveGroup(7,1);

Polynomial with Galois group C7 over ℚ
actionf(x)Disc(f)
7T1x7-x6-12x5+7x4+28x3-14x2-9x-1172·296

Matrix representation of C7 in GL1(𝔽29) generated by

7
G:=sub<GL(1,GF(29))| [7] >;

C7 in GAP, Magma, Sage, TeX

C_7
% in TeX

G:=Group("C7");
// GroupNames label

G:=SmallGroup(7,1);
// by ID

G=gap.SmallGroup(7,1);
# by ID

G:=PCGroup([1,-7]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^7=1>;
// generators/relations

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