direct product, cyclic, abelian, monomial
Aliases: C20, also denoted Z20, SmallGroup(20,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C20 |
| C1 — C20 |
| C1 — C20 |
Generators and relations for C20
G = < a | a20=1 >
Character table of C20
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 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 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -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 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | i | -i | i | -i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | i | -i | i | -i | i | i | linear of order 4 |
| ρ5 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | ζ5 | linear of order 5 |
| ρ6 | 1 | -1 | -i | i | ζ52 | ζ53 | ζ54 | ζ5 | -ζ52 | -ζ54 | -ζ53 | -ζ5 | ζ4ζ54 | ζ4ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ54 | ζ43ζ5 | linear of order 20 faithful |
| ρ7 | 1 | 1 | -1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | -ζ54 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | -ζ5 | linear of order 10 |
| ρ8 | 1 | -1 | i | -i | ζ52 | ζ53 | ζ54 | ζ5 | -ζ52 | -ζ54 | -ζ53 | -ζ5 | ζ43ζ54 | ζ43ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ54 | ζ4ζ5 | linear of order 20 faithful |
| ρ9 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | ζ52 | linear of order 5 |
| ρ10 | 1 | -1 | -i | i | ζ54 | ζ5 | ζ53 | ζ52 | -ζ54 | -ζ53 | -ζ5 | -ζ52 | ζ4ζ53 | ζ4ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ53 | ζ43ζ52 | linear of order 20 faithful |
| ρ11 | 1 | 1 | -1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | -ζ53 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | -ζ52 | linear of order 10 |
| ρ12 | 1 | -1 | i | -i | ζ54 | ζ5 | ζ53 | ζ52 | -ζ54 | -ζ53 | -ζ5 | -ζ52 | ζ43ζ53 | ζ43ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ53 | ζ4ζ52 | linear of order 20 faithful |
| ρ13 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | ζ53 | linear of order 5 |
| ρ14 | 1 | -1 | -i | i | ζ5 | ζ54 | ζ52 | ζ53 | -ζ5 | -ζ52 | -ζ54 | -ζ53 | ζ4ζ52 | ζ4ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ52 | ζ43ζ53 | linear of order 20 faithful |
| ρ15 | 1 | 1 | -1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | -ζ52 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | -ζ53 | linear of order 10 |
| ρ16 | 1 | -1 | i | -i | ζ5 | ζ54 | ζ52 | ζ53 | -ζ5 | -ζ52 | -ζ54 | -ζ53 | ζ43ζ52 | ζ43ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ52 | ζ4ζ53 | linear of order 20 faithful |
| ρ17 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | ζ54 | linear of order 5 |
| ρ18 | 1 | -1 | -i | i | ζ53 | ζ52 | ζ5 | ζ54 | -ζ53 | -ζ5 | -ζ52 | -ζ54 | ζ4ζ5 | ζ4ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ5 | ζ43ζ54 | linear of order 20 faithful |
| ρ19 | 1 | 1 | -1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | -ζ5 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | -ζ54 | linear of order 10 |
| ρ20 | 1 | -1 | i | -i | ζ53 | ζ52 | ζ5 | ζ54 | -ζ53 | -ζ5 | -ζ52 | -ζ54 | ζ43ζ5 | ζ43ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ5 | ζ4ζ54 | linear of order 20 faithful |
►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);
C20 is a maximal subgroup of
C5⋊2C8 Dic10 D20 C11⋊C20 2- 1+4.C10
C20 is a maximal quotient of C11⋊C20
| action | f(x) | Disc(f) |
|---|---|---|
| 20T1 | x20+x15+x10+x5+1 | 535 |
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
Export
Subgroup lattice of C20 in TeX
Character table of C20 in TeX