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## G = C28order 28 = 22·7

### Cyclic group

Aliases: C28, also denoted Z28, SmallGroup(28,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28
 Chief series C1 — C2 — C14 — C28
 Lower central C1 — C28
 Upper central C1 — C28

Generators and relations for C28
G = < a | a28=1 >

Character table of C28

 class 1 2 4A 4B 7A 7B 7C 7D 7E 7F 14A 14B 14C 14D 14E 14F 28A 28B 28C 28D 28E 28F 28G 28H 28I 28J 28K 28L size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 i i -i i -i i -i i -i i -i -i linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -i -i i -i i -i i -i i -i i i linear of order 4 ρ5 1 1 1 1 ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 ζ73 ζ75 ζ72 ζ76 ζ74 ζ7 ζ76 ζ7 ζ72 ζ72 ζ73 ζ73 ζ74 ζ74 ζ75 ζ75 ζ76 ζ7 linear of order 7 ρ6 1 -1 -i i ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 -ζ73 -ζ75 -ζ72 -ζ76 -ζ74 -ζ7 ζ4ζ76 ζ4ζ7 ζ43ζ72 ζ4ζ72 ζ43ζ73 ζ4ζ73 ζ43ζ74 ζ4ζ74 ζ43ζ75 ζ4ζ75 ζ43ζ76 ζ43ζ7 linear of order 28 faithful ρ7 1 1 -1 -1 ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 ζ73 ζ75 ζ72 ζ76 ζ74 ζ7 -ζ76 -ζ7 -ζ72 -ζ72 -ζ73 -ζ73 -ζ74 -ζ74 -ζ75 -ζ75 -ζ76 -ζ7 linear of order 14 ρ8 1 -1 i -i ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 -ζ73 -ζ75 -ζ72 -ζ76 -ζ74 -ζ7 ζ43ζ76 ζ43ζ7 ζ4ζ72 ζ43ζ72 ζ4ζ73 ζ43ζ73 ζ4ζ74 ζ43ζ74 ζ4ζ75 ζ43ζ75 ζ4ζ76 ζ4ζ7 linear of order 28 faithful ρ9 1 1 1 1 ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 ζ76 ζ73 ζ74 ζ75 ζ7 ζ72 ζ75 ζ72 ζ74 ζ74 ζ76 ζ76 ζ7 ζ7 ζ73 ζ73 ζ75 ζ72 linear of order 7 ρ10 1 -1 -i i ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 -ζ76 -ζ73 -ζ74 -ζ75 -ζ7 -ζ72 ζ4ζ75 ζ4ζ72 ζ43ζ74 ζ4ζ74 ζ43ζ76 ζ4ζ76 ζ43ζ7 ζ4ζ7 ζ43ζ73 ζ4ζ73 ζ43ζ75 ζ43ζ72 linear of order 28 faithful ρ11 1 1 -1 -1 ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 ζ76 ζ73 ζ74 ζ75 ζ7 ζ72 -ζ75 -ζ72 -ζ74 -ζ74 -ζ76 -ζ76 -ζ7 -ζ7 -ζ73 -ζ73 -ζ75 -ζ72 linear of order 14 ρ12 1 -1 i -i ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 -ζ76 -ζ73 -ζ74 -ζ75 -ζ7 -ζ72 ζ43ζ75 ζ43ζ72 ζ4ζ74 ζ43ζ74 ζ4ζ76 ζ43ζ76 ζ4ζ7 ζ43ζ7 ζ4ζ73 ζ43ζ73 ζ4ζ75 ζ4ζ72 linear of order 28 faithful ρ13 1 1 1 1 ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 ζ72 ζ7 ζ76 ζ74 ζ75 ζ73 ζ74 ζ73 ζ76 ζ76 ζ72 ζ72 ζ75 ζ75 ζ7 ζ7 ζ74 ζ73 linear of order 7 ρ14 1 -1 -i i ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 -ζ72 -ζ7 -ζ76 -ζ74 -ζ75 -ζ73 ζ4ζ74 ζ4ζ73 ζ43ζ76 ζ4ζ76 ζ43ζ72 ζ4ζ72 ζ43ζ75 ζ4ζ75 ζ43ζ7 ζ4ζ7 ζ43ζ74 ζ43ζ73 linear of order 28 faithful ρ15 1 1 -1 -1 ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 ζ72 ζ7 ζ76 ζ74 ζ75 ζ73 -ζ74 -ζ73 -ζ76 -ζ76 -ζ72 -ζ72 -ζ75 -ζ75 -ζ7 -ζ7 -ζ74 -ζ73 linear of order 14 ρ16 1 -1 i -i ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 -ζ72 -ζ7 -ζ76 -ζ74 -ζ75 -ζ73 ζ43ζ74 ζ43ζ73 ζ4ζ76 ζ43ζ76 ζ4ζ72 ζ43ζ72 ζ4ζ75 ζ43ζ75 ζ4ζ7 ζ43ζ7 ζ4ζ74 ζ4ζ73 linear of order 28 faithful ρ17 1 1 1 1 ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 ζ75 ζ76 ζ7 ζ73 ζ72 ζ74 ζ73 ζ74 ζ7 ζ7 ζ75 ζ75 ζ72 ζ72 ζ76 ζ76 ζ73 ζ74 linear of order 7 ρ18 1 -1 -i i ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 -ζ75 -ζ76 -ζ7 -ζ73 -ζ72 -ζ74 ζ4ζ73 ζ4ζ74 ζ43ζ7 ζ4ζ7 ζ43ζ75 ζ4ζ75 ζ43ζ72 ζ4ζ72 ζ43ζ76 ζ4ζ76 ζ43ζ73 ζ43ζ74 linear of order 28 faithful ρ19 1 1 -1 -1 ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 ζ75 ζ76 ζ7 ζ73 ζ72 ζ74 -ζ73 -ζ74 -ζ7 -ζ7 -ζ75 -ζ75 -ζ72 -ζ72 -ζ76 -ζ76 -ζ73 -ζ74 linear of order 14 ρ20 1 -1 i -i ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 -ζ75 -ζ76 -ζ7 -ζ73 -ζ72 -ζ74 ζ43ζ73 ζ43ζ74 ζ4ζ7 ζ43ζ7 ζ4ζ75 ζ43ζ75 ζ4ζ72 ζ43ζ72 ζ4ζ76 ζ43ζ76 ζ4ζ73 ζ4ζ74 linear of order 28 faithful ρ21 1 1 1 1 ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 ζ7 ζ74 ζ73 ζ72 ζ76 ζ75 ζ72 ζ75 ζ73 ζ73 ζ7 ζ7 ζ76 ζ76 ζ74 ζ74 ζ72 ζ75 linear of order 7 ρ22 1 -1 -i i ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 -ζ7 -ζ74 -ζ73 -ζ72 -ζ76 -ζ75 ζ4ζ72 ζ4ζ75 ζ43ζ73 ζ4ζ73 ζ43ζ7 ζ4ζ7 ζ43ζ76 ζ4ζ76 ζ43ζ74 ζ4ζ74 ζ43ζ72 ζ43ζ75 linear of order 28 faithful ρ23 1 1 -1 -1 ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 ζ7 ζ74 ζ73 ζ72 ζ76 ζ75 -ζ72 -ζ75 -ζ73 -ζ73 -ζ7 -ζ7 -ζ76 -ζ76 -ζ74 -ζ74 -ζ72 -ζ75 linear of order 14 ρ24 1 -1 i -i ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 -ζ7 -ζ74 -ζ73 -ζ72 -ζ76 -ζ75 ζ43ζ72 ζ43ζ75 ζ4ζ73 ζ43ζ73 ζ4ζ7 ζ43ζ7 ζ4ζ76 ζ43ζ76 ζ4ζ74 ζ43ζ74 ζ4ζ72 ζ4ζ75 linear of order 28 faithful ρ25 1 1 1 1 ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 ζ74 ζ72 ζ75 ζ7 ζ73 ζ76 ζ7 ζ76 ζ75 ζ75 ζ74 ζ74 ζ73 ζ73 ζ72 ζ72 ζ7 ζ76 linear of order 7 ρ26 1 -1 -i i ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 -ζ74 -ζ72 -ζ75 -ζ7 -ζ73 -ζ76 ζ4ζ7 ζ4ζ76 ζ43ζ75 ζ4ζ75 ζ43ζ74 ζ4ζ74 ζ43ζ73 ζ4ζ73 ζ43ζ72 ζ4ζ72 ζ43ζ7 ζ43ζ76 linear of order 28 faithful ρ27 1 1 -1 -1 ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 ζ74 ζ72 ζ75 ζ7 ζ73 ζ76 -ζ7 -ζ76 -ζ75 -ζ75 -ζ74 -ζ74 -ζ73 -ζ73 -ζ72 -ζ72 -ζ7 -ζ76 linear of order 14 ρ28 1 -1 i -i ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 -ζ74 -ζ72 -ζ75 -ζ7 -ζ73 -ζ76 ζ43ζ7 ζ43ζ76 ζ4ζ75 ζ43ζ75 ζ4ζ74 ζ43ζ74 ζ4ζ73 ζ43ζ73 ζ4ζ72 ζ43ζ72 ζ4ζ7 ζ4ζ76 linear of order 28 faithful

Permutation representations of C28
Regular action on 28 points - transitive group 28T1
Generators in S28
`(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)`

`G:=sub<Sym(28)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)]])`

`G:=TransitiveGroup(28,1);`

C28 is a maximal subgroup of   C7⋊C8  Dic14  D28

Polynomial with Galois group C28 over ℚ
actionf(x)Disc(f)
28T1x28+x27+x26+x25+x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+12927

Matrix representation of C28 in GL1(𝔽29) generated by

 26
`G:=sub<GL(1,GF(29))| [26] >;`

C28 in GAP, Magma, Sage, TeX

`C_{28}`
`% in TeX`

`G:=Group("C28");`
`// GroupNames label`

`G:=SmallGroup(28,2);`
`// by ID`

`G=gap.SmallGroup(28,2);`
`# by ID`

`G:=PCGroup([3,-2,-7,-2,42]);`
`// Polycyclic`

`G:=Group<a|a^28=1>;`
`// generators/relations`

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