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G = C15order 15 = 3·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C15, also denoted Z15, SmallGroup(15,1)

Series: Derived Chief Lower central Upper central

C1 — C15
C1C5 — C15
C1 — C15
C1 — C15

Generators and relations for C15
 G = < a | a15=1 >


Character table of C15

 class 13A3B5A5B5C5D15A15B15C15D15E15F15G15H
 size 111111111111111
ρ1111111111111111    trivial
ρ21ζ32ζ31111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ31ζ3ζ321111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ4111ζ53ζ54ζ5ζ52ζ54ζ53ζ54ζ5ζ5ζ52ζ53ζ52    linear of order 5
ρ51ζ32ζ2ζ53ζ54ζ5ζ52ζ32ζ54ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ3ζ52    linear of order 15 faithful
ρ61ζ2ζ32ζ53ζ54ζ5ζ52ζ3ζ54ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ32ζ52    linear of order 15 faithful
ρ7111ζ5ζ53ζ52ζ54ζ53ζ5ζ53ζ52ζ52ζ54ζ5ζ54    linear of order 5
ρ81ζ32ζ2ζ5ζ53ζ52ζ54ζ32ζ53ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ3ζ54    linear of order 15 faithful
ρ91ζ2ζ32ζ5ζ53ζ52ζ54ζ3ζ53ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ32ζ54    linear of order 15 faithful
ρ10111ζ54ζ52ζ53ζ5ζ52ζ54ζ52ζ53ζ53ζ5ζ54ζ5    linear of order 5
ρ111ζ32ζ2ζ54ζ52ζ53ζ5ζ32ζ52ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ3ζ5    linear of order 15 faithful
ρ121ζ2ζ32ζ54ζ52ζ53ζ5ζ3ζ52ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ32ζ5    linear of order 15 faithful
ρ13111ζ52ζ5ζ54ζ53ζ5ζ52ζ5ζ54ζ54ζ53ζ52ζ53    linear of order 5
ρ141ζ32ζ2ζ52ζ5ζ54ζ53ζ32ζ5ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ3ζ53    linear of order 15 faithful
ρ151ζ2ζ32ζ52ζ5ζ54ζ53ζ3ζ5ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ32ζ53    linear of order 15 faithful

Permutation representations of C15
Regular action on 15 points - transitive group 15T1
Generators in S15
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)

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

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

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

G:=TransitiveGroup(15,1);

Polynomial with Galois group C15 over ℚ
actionf(x)Disc(f)
15T1x15-2x14-23x13+42x12+182x11-300x10-614x9+885x8+918x7-1112x6-525x5+508x4+60x3-65x2+x+1710·1112·432·12312·15832·543612

Matrix representation of C15 in GL1(𝔽31) generated by

19
G:=sub<GL(1,GF(31))| [19] >;

C15 in GAP, Magma, Sage, TeX

C_{15}
% in TeX

G:=Group("C15");
// GroupNames label

G:=SmallGroup(15,1);
// by ID

G=gap.SmallGroup(15,1);
# by ID

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

G:=Group<a|a^15=1>;
// generators/relations

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