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G = C10order 10 = 2·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C10, also denoted Z10, SmallGroup(10,2)

Series: Derived Chief Lower central Upper central

C1 — C10
C1C5 — C10
C1 — C10
C1 — C10

Generators and relations for C10
 G = < a | a10=1 >


Character table of C10

 class 125A5B5C5D10A10B10C10D
 size 1111111111
ρ11111111111    trivial
ρ21-11111-1-1-1-1    linear of order 2
ρ311ζ52ζ53ζ54ζ5ζ54ζ52ζ53ζ5    linear of order 5
ρ41-1ζ52ζ53ζ54ζ55452535    linear of order 10 faithful
ρ511ζ54ζ5ζ53ζ52ζ53ζ54ζ5ζ52    linear of order 5
ρ61-1ζ54ζ5ζ53ζ525354552    linear of order 10 faithful
ρ711ζ5ζ54ζ52ζ53ζ52ζ5ζ54ζ53    linear of order 5
ρ81-1ζ5ζ54ζ52ζ535255453    linear of order 10 faithful
ρ911ζ53ζ52ζ5ζ54ζ5ζ53ζ52ζ54    linear of order 5
ρ101-1ζ53ζ52ζ5ζ545535254    linear of order 10 faithful

Permutation representations of C10
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);

Polynomial with Galois group C10 over ℚ
actionf(x)Disc(f)
10T1x10+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

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