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G = C7order 7

Cyclic group

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C7
 Chief series C1 — C7
 Lower central C1 — C7
 Upper central C1 — C7
 Jennings C1 — C7

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

Character table of C7

 class 1 7A 7B 7C 7D 7E 7F size 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 linear of order 7 faithful ρ3 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 linear of order 7 faithful ρ4 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 linear of order 7 faithful ρ5 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 linear of order 7 faithful ρ6 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 linear of order 7 faithful ρ7 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 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);`

C7 is a maximal subgroup of   D7  C7⋊C3  C49  F8  C29⋊C7  C43⋊C7  C71⋊C7
C7 is a maximal quotient of   C49  F8  C29⋊C7  C43⋊C7  C71⋊C7

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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