direct product, cyclic, abelian, monomial
Aliases: C18, also denoted Z18, SmallGroup(18,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C18 |
| C1 — C18 |
| C1 — C18 |
Generators and relations for C18
G = < a | a18=1 >
Character table of C18
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 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 | trivial |
| ρ2 | 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 | ζ32 | ζ3 | ζ3 | ζ32 | ζ95 | ζ98 | ζ92 | ζ97 | ζ9 | ζ94 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | linear of order 9 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ95 | ζ98 | ζ92 | ζ97 | ζ9 | ζ94 | -ζ92 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | linear of order 18 faithful |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ9 | ζ97 | ζ94 | ζ95 | ζ92 | ζ98 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | linear of order 9 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ9 | ζ97 | ζ94 | ζ95 | ζ92 | ζ98 | -ζ94 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | linear of order 18 faithful |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ92 | ζ95 | ζ98 | ζ9 | ζ94 | ζ97 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | linear of order 9 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ92 | ζ95 | ζ98 | ζ9 | ζ94 | ζ97 | -ζ98 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | linear of order 18 faithful |
| ρ11 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ97 | ζ94 | ζ9 | ζ98 | ζ95 | ζ92 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | linear of order 9 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ97 | ζ94 | ζ9 | ζ98 | ζ95 | ζ92 | -ζ9 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | linear of order 18 faithful |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ14 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ98 | ζ92 | ζ95 | ζ94 | ζ97 | ζ9 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | linear of order 9 |
| ρ16 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ98 | ζ92 | ζ95 | ζ94 | ζ97 | ζ9 | -ζ95 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | linear of order 18 faithful |
| ρ17 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ94 | ζ9 | ζ97 | ζ92 | ζ98 | ζ95 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | linear of order 9 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ94 | ζ9 | ζ97 | ζ92 | ζ98 | ζ95 | -ζ97 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | linear of order 18 faithful |
►Regular action on 18 points - transitive group 18T1
Generators in S18
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)]])
G:=TransitiveGroup(18,1);
C18 is a maximal subgroup of
Dic9 Q8⋊C9 C7⋊C18 C13⋊C18 C57.C6 F19 C52⋊C18
C18 is a maximal quotient of C7⋊C18 C13⋊C18 C57.C6 F19 C52⋊C18
| action | f(x) | Disc(f) |
|---|---|---|
| 18T1 | x18+x9+1 | -345 |
Matrix representation of C18 ►in GL1(𝔽19) generated by
| 14 |
G:=sub<GL(1,GF(19))| [14] >;
C18 in GAP, Magma, Sage, TeX
C_{18} % in TeX
G:=Group("C18"); // GroupNames label
G:=SmallGroup(18,2);
// by ID
G=gap.SmallGroup(18,2);
# by ID
G:=PCGroup([3,-2,-3,-3,22]);
// Polycyclic
G:=Group<a|a^18=1>;
// generators/relations
Export
Subgroup lattice of C18 in TeX
Character table of C18 in TeX