direct product, cyclic, abelian, monomial
Aliases: C12, also denoted Z12, SmallGroup(12,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C12 |
| C1 — C12 |
| 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 |
►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 C52⋊2C12 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
C52⋊2C12 C52⋊C12
Cp⋊C12, p=1 mod 3: C7⋊C12 C26.C6 F13 C19⋊C12 C31⋊C12 C74.C6 C37⋊C12 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 12T1 | x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | 1311 |
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
Export
Subgroup lattice of C12 in TeX
Character table of C12 in TeX