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## G = C12order 12 = 22·3

### Cyclic group

Aliases: C12, also denoted Z12, SmallGroup(12,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12
 Chief series C1 — C2 — C6 — C12
 Lower central C1 — C12
 Upper central C1 — C12

Generators and relations for C12
G = < a | a12=1 >

Character table of C12

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D size 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 trivial ρ2 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 1 -i i -1 -1 i i -i -i linear of order 4 ρ4 1 -1 1 1 i -i -1 -1 -i -i i i linear of order 4 ρ5 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 -1 ζ3 ζ32 -i i ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 linear of order 12 faithful ρ7 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ8 1 -1 ζ3 ζ32 i -i ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 linear of order 12 faithful ρ9 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ10 1 -1 ζ32 ζ3 -i i ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 linear of order 12 faithful ρ11 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ12 1 -1 ζ32 ζ3 i -i ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 linear of order 12 faithful

Permutation representations of C12
Regular action on 12 points - transitive group 12T1
Generators in S12
`(1 2 3 4 5 6 7 8 9 10 11 12)`

`G:=sub<Sym(12)| (1,2,3,4,5,6,7,8,9,10,11,12)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12)])`

`G:=TransitiveGroup(12,1);`

C12 is a maximal subgroup of
C3⋊C8  Dic6  D12  C4.A4  He3⋊C4  C522C12  C52⋊C12
Cp⋊C12, p=1 mod 3: C7⋊C12  C26.C6  F13  C19⋊C12  C31⋊C12  C74.C6  C37⋊C12 ...
C12 is a maximal quotient of
C522C12  C52⋊C12
Cp⋊C12, p=1 mod 3: C7⋊C12  C26.C6  F13  C19⋊C12  C31⋊C12  C74.C6  C37⋊C12 ...

Polynomial with Galois group C12 over ℚ
actionf(x)Disc(f)
12T1x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+11311

Matrix representation of C12 in GL1(𝔽13) generated by

 6
`G:=sub<GL(1,GF(13))| [6] >;`

C12 in GAP, Magma, Sage, TeX

`C_{12}`
`% in TeX`

`G:=Group("C12");`
`// GroupNames label`

`G:=SmallGroup(12,2);`
`// by ID`

`G=gap.SmallGroup(12,2);`
`# by ID`

`G:=PCGroup([3,-2,-3,-2,18]);`
`// Polycyclic`

`G:=Group<a|a^12=1>;`
`// generators/relations`

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