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G = D15order 30 = 2·3·5

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D15, C5⋊S3, C3⋊D5, C151C2, sometimes denoted D30 or Dih15 or Dih30, SmallGroup(30,3)

Series: Derived Chief Lower central Upper central

C1C15 — D15
C1C5C15 — D15
C15 — D15
C1

Generators and relations for D15
 G = < a,b | a15=b2=1, bab=a-1 >

15C2
5S3
3D5

Character table of D15

 class 1235A5B15A15B15C15D
 size 1152222222
ρ1111111111    trivial
ρ21-11111111    linear of order 2
ρ320-122-1-1-1-1    orthogonal lifted from S3
ρ4202-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ5202-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ620-1-1-5/2-1+5/23ζ533ζ5253ζ3ζ533ζ5252ζ3ζ543ζ55ζ32ζ5432ζ55    orthogonal faithful
ρ720-1-1+5/2-1-5/2ζ32ζ5432ζ55ζ3ζ543ζ553ζ533ζ5253ζ3ζ533ζ5252    orthogonal faithful
ρ820-1-1-5/2-1+5/2ζ3ζ533ζ52523ζ533ζ5253ζ32ζ5432ζ55ζ3ζ543ζ55    orthogonal faithful
ρ920-1-1+5/2-1-5/2ζ3ζ543ζ55ζ32ζ5432ζ55ζ3ζ533ζ52523ζ533ζ5253    orthogonal faithful

Permutation representations of D15
On 15 points - transitive group 15T2
Generators in S15
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)

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

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

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

G:=TransitiveGroup(15,2);

Regular action on 30 points - transitive group 30T3
Generators in S30
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)

G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,30),(14,29),(15,28)])

G:=TransitiveGroup(30,3);

Polynomial with Galois group D15 over ℚ
actionf(x)Disc(f)
15T2x15-x14+17x13+138x12-188x11+875x10+6364x9-23852x8-50961x7+192039x6+95751x5-707345x4+775903x3-478177x2+162804x-22825-54·192·2310·477·836·2292·22872·38772·840672·11085072·12519112·1300732692

Matrix representation of D15 in GL2(𝔽29) generated by

154
2528
,
280
41
G:=sub<GL(2,GF(29))| [15,25,4,28],[28,4,0,1] >;

D15 in GAP, Magma, Sage, TeX

D_{15}
% in TeX

G:=Group("D15");
// GroupNames label

G:=SmallGroup(30,3);
// by ID

G=gap.SmallGroup(30,3);
# by ID

G:=PCGroup([3,-2,-3,-5,25,218]);
// Polycyclic

G:=Group<a,b|a^15=b^2=1,b*a*b=a^-1>;
// generators/relations

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