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G = C18order 18 = 2·32

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C18, also denoted Z18, SmallGroup(18,2)

Series: Derived Chief Lower central Upper central

C1 — C18
C1C3C9 — C18
C1 — C18
C1 — C18

Generators and relations for C18
 G = < a | a18=1 >


Character table of C18

 class 123A3B6A6B9A9B9C9D9E9F18A18B18C18D18E18F
 size 111111111111111111
ρ1111111111111111111    trivial
ρ21-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ32ζ95ζ98ζ92ζ97ζ9ζ94ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ41-1ζ32ζ3ζ65ζ6ζ95ζ98ζ92ζ97ζ9ζ9492994959897    linear of order 18 faithful
ρ511ζ3ζ32ζ32ζ3ζ9ζ97ζ94ζ95ζ92ζ98ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ61-1ζ3ζ32ζ6ζ65ζ9ζ97ζ94ζ95ζ92ζ9894929899795    linear of order 18 faithful
ρ7111111ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ81-111-1-1ζ32ζ32ζ32ζ3ζ3ζ3ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ911ζ32ζ3ζ3ζ32ζ92ζ95ζ98ζ9ζ94ζ97ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ101-1ζ32ζ3ζ65ζ6ζ92ζ95ζ98ζ9ζ94ζ9798949792959    linear of order 18 faithful
ρ1111ζ3ζ32ζ32ζ3ζ97ζ94ζ9ζ98ζ95ζ92ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ121-1ζ3ζ32ζ6ζ65ζ97ζ94ζ9ζ98ζ95ζ9299592979498    linear of order 18 faithful
ρ13111111ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ141-111-1-1ζ3ζ3ζ3ζ32ζ32ζ32ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ1511ζ32ζ3ζ3ζ32ζ98ζ92ζ95ζ94ζ97ζ9ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ161-1ζ32ζ3ζ65ζ6ζ98ζ92ζ95ζ94ζ97ζ995979989294    linear of order 18 faithful
ρ1711ζ3ζ32ζ32ζ3ζ94ζ9ζ97ζ92ζ98ζ95ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ181-1ζ3ζ32ζ6ζ65ζ94ζ9ζ97ζ92ζ98ζ9597989594992    linear of order 18 faithful

Permutation representations of C18
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);

Polynomial with Galois group C18 over ℚ
actionf(x)Disc(f)
18T1x18+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

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