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G = C20order 20 = 22·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C20, also denoted Z20, SmallGroup(20,2)

Series: Derived Chief Lower central Upper central

C1 — C20
C1C2C10 — C20
C1 — C20
C1 — C20

Generators and relations for C20
 G = < a | a20=1 >


Character table of C20

 class 124A4B5A5B5C5D10A10B10C10D20A20B20C20D20E20F20G20H
 size 11111111111111111111
ρ111111111111111111111    trivial
ρ211-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1-ii1111-1-1-1-1ii-ii-ii-i-i    linear of order 4
ρ41-1i-i1111-1-1-1-1-i-ii-ii-iii    linear of order 4
ρ51111ζ52ζ53ζ54ζ5ζ52ζ54ζ53ζ5ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ5    linear of order 5
ρ61-1ζ2ζ2ζ52ζ53ζ54ζ55254535ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ5    linear of order 20 faithful
ρ711-1-1ζ52ζ53ζ54ζ5ζ52ζ54ζ53ζ554552525353545    linear of order 10
ρ81-1ζ2ζ2ζ52ζ53ζ54ζ55254535ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ5    linear of order 20 faithful
ρ91111ζ54ζ5ζ53ζ52ζ54ζ53ζ5ζ52ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ52    linear of order 5
ρ101-1ζ2ζ2ζ54ζ5ζ53ζ525453552ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ52    linear of order 20 faithful
ρ1111-1-1ζ54ζ5ζ53ζ52ζ54ζ53ζ5ζ5253525454555352    linear of order 10
ρ121-1ζ2ζ2ζ54ζ5ζ53ζ525453552ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ52    linear of order 20 faithful
ρ131111ζ5ζ54ζ52ζ53ζ5ζ52ζ54ζ53ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ53    linear of order 5
ρ141-1ζ2ζ2ζ5ζ54ζ52ζ535525453ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ53    linear of order 20 faithful
ρ1511-1-1ζ5ζ54ζ52ζ53ζ5ζ52ζ54ζ5352535554545253    linear of order 10
ρ161-1ζ2ζ2ζ5ζ54ζ52ζ535525453ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ53    linear of order 20 faithful
ρ171111ζ53ζ52ζ5ζ54ζ53ζ5ζ52ζ54ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ54    linear of order 5
ρ181-1ζ2ζ2ζ53ζ52ζ5ζ545355254ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ54    linear of order 20 faithful
ρ1911-1-1ζ53ζ52ζ5ζ54ζ53ζ5ζ52ζ5455453535252554    linear of order 10
ρ201-1ζ2ζ2ζ53ζ52ζ5ζ545355254ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ54    linear of order 20 faithful

Permutation representations of C20
Regular action on 20 points - transitive group 20T1
Generators in S20
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)])

G:=TransitiveGroup(20,1);

Polynomial with Galois group C20 over ℚ
actionf(x)Disc(f)
20T1x20+x15+x10+x5+1535

Matrix representation of C20 in GL1(𝔽41) generated by

21
G:=sub<GL(1,GF(41))| [21] >;

C20 in GAP, Magma, Sage, TeX

C_{20}
% in TeX

G:=Group("C20");
// GroupNames label

G:=SmallGroup(20,2);
// by ID

G=gap.SmallGroup(20,2);
# by ID

G:=PCGroup([3,-2,-5,-2,30]);
// Polycyclic

G:=Group<a|a^20=1>;
// generators/relations

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