direct product, cyclic, abelian, monomial
Aliases: C10, also denoted Z10, SmallGroup(10,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C10 |
| C1 — C10 |
| C1 — C10 |
Generators and relations for C10
G = < a | a10=1 >
Character table of C10
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ52 | ζ53 | ζ5 | linear of order 5 |
| ρ4 | 1 | -1 | ζ52 | ζ53 | ζ54 | ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | linear of order 10 faithful |
| ρ5 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ54 | ζ5 | ζ52 | linear of order 5 |
| ρ6 | 1 | -1 | ζ54 | ζ5 | ζ53 | ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | linear of order 10 faithful |
| ρ7 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ5 | ζ54 | ζ53 | linear of order 5 |
| ρ8 | 1 | -1 | ζ5 | ζ54 | ζ52 | ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | linear of order 10 faithful |
| ρ9 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ53 | ζ52 | ζ54 | linear of order 5 |
| ρ10 | 1 | -1 | ζ53 | ζ52 | ζ5 | ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | linear of order 10 faithful |
►Regular action on 10 points - transitive group 10T1
Generators in S10
(1 2 3 4 5 6 7 8 9 10)
G:=sub<Sym(10)| (1,2,3,4,5,6,7,8,9,10)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10)]])
G:=TransitiveGroup(10,1);
C10 is a maximal subgroup of
Dic5 F11 2- 1+4⋊C5 C31⋊C10 C41⋊C10
C10 is a maximal quotient of F11 C31⋊C10 C41⋊C10
| action | f(x) | Disc(f) |
|---|---|---|
| 10T1 | x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | -119 |
Matrix representation of C10 ►in GL1(𝔽11) generated by
| 6 |
G:=sub<GL(1,GF(11))| [6] >;
C10 in GAP, Magma, Sage, TeX
C_{10} % in TeX
G:=Group("C10"); // GroupNames label
G:=SmallGroup(10,2);
// by ID
G=gap.SmallGroup(10,2);
# by ID
G:=PCGroup([2,-2,-5]);
// Polycyclic
G:=Group<a|a^10=1>;
// generators/relations
Export
Subgroup lattice of C10 in TeX
Character table of C10 in TeX