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G = C12order 12 = 22·3

Cyclic group

direct product, cyclic, abelian, monomial

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

Series: Derived Chief Lower central Upper central

C1 — C12
C1C2C6 — C12
C1 — C12
C1 — C12

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


Character table of C12

 class 123A3B4A4B6A6B12A12B12C12D
 size 111111111111
ρ1111111111111    trivial
ρ21111-1-111-1-1-1-1    linear of order 2
ρ31-111-ii-1-1ii-i-i    linear of order 4
ρ41-111i-i-1-1-i-iii    linear of order 4
ρ511ζ3ζ3211ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ61-1ζ62ζ32ζ2ζ2ζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    linear of order 12 faithful
ρ711ζ3ζ32-1-1ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ81-1ζ62ζ32ζ2ζ2ζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    linear of order 12 faithful
ρ911ζ32ζ311ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ101-1ζ32ζ62ζ2ζ2ζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    linear of order 12 faithful
ρ1111ζ32ζ3-1-1ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ121-1ζ32ζ62ζ2ζ2ζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    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);

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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