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G = D30order 60 = 22·3·5

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D30, C2×D15, C6⋊D5, C10⋊S3, C52D6, C32D10, C301C2, C152C22, sometimes denoted D60 or Dih30 or Dih60, SmallGroup(60,12)

Series: Derived Chief Lower central Upper central

C1C15 — D30
C1C5C15D15 — D30
C15 — D30
C1C2

Generators and relations for D30
 G = < a,b | a30=b2=1, bab=a-1 >

15C2
15C2
15C22
5S3
5S3
3D5
3D5
5D6
3D10

Character table of D30

 class 12A2B2C35A5B610A10B15A15B15C15D30A30B30C30D
 size 11151522222222222222
ρ1111111111111111111    trivial
ρ21-1-11111-1-1-11111-1-1-1-1    linear of order 2
ρ311-1-111111111111111    linear of order 2
ρ41-11-1111-1-1-11111-1-1-1-1    linear of order 2
ρ52-200-1221-2-2-1-1-1-11111    orthogonal lifted from D6
ρ62200-122-122-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ72-2002-1-5/2-1+5/2-21-5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2    orthogonal lifted from D10
ρ82-2002-1+5/2-1-5/2-21+5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2    orthogonal lifted from D10
ρ922002-1-5/2-1+5/22-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ1022002-1+5/2-1-5/22-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ112200-1-1-5/2-1+5/2-1-1+5/2-1-5/232ζ5432ζ5543ζ533ζ52533ζ543ζ554ζ3ζ533ζ52523ζ543ζ554ζ3ζ533ζ52523ζ533ζ525332ζ5432ζ554    orthogonal lifted from D15
ρ122200-1-1-5/2-1+5/2-1-1+5/2-1-5/23ζ543ζ554ζ3ζ533ζ525232ζ5432ζ5543ζ533ζ525332ζ5432ζ5543ζ533ζ5253ζ3ζ533ζ52523ζ543ζ554    orthogonal lifted from D15
ρ132200-1-1+5/2-1-5/2-1-1-5/2-1+5/23ζ533ζ52533ζ543ζ554ζ3ζ533ζ525232ζ5432ζ554ζ3ζ533ζ525232ζ5432ζ5543ζ543ζ5543ζ533ζ5253    orthogonal lifted from D15
ρ142-200-1-1+5/2-1-5/211+5/21-5/23ζ533ζ52533ζ543ζ554ζ3ζ533ζ525232ζ5432ζ5543ζ533ζ5252ζ32ζ5432ζ554ζ3ζ543ζ554ζ3ζ533ζ5253    orthogonal faithful
ρ152-200-1-1-5/2-1+5/211-5/21+5/232ζ5432ζ5543ζ533ζ52533ζ543ζ554ζ3ζ533ζ5252ζ3ζ543ζ5543ζ533ζ5252ζ3ζ533ζ5253ζ32ζ5432ζ554    orthogonal faithful
ρ162-200-1-1-5/2-1+5/211-5/21+5/23ζ543ζ554ζ3ζ533ζ525232ζ5432ζ5543ζ533ζ5253ζ32ζ5432ζ554ζ3ζ533ζ52533ζ533ζ5252ζ3ζ543ζ554    orthogonal faithful
ρ172-200-1-1+5/2-1-5/211+5/21-5/2ζ3ζ533ζ525232ζ5432ζ5543ζ533ζ52533ζ543ζ554ζ3ζ533ζ5253ζ3ζ543ζ554ζ32ζ5432ζ5543ζ533ζ5252    orthogonal faithful
ρ182200-1-1+5/2-1-5/2-1-1-5/2-1+5/2ζ3ζ533ζ525232ζ5432ζ5543ζ533ζ52533ζ543ζ5543ζ533ζ52533ζ543ζ55432ζ5432ζ554ζ3ζ533ζ5252    orthogonal lifted from D15

Permutation representations of D30
On 30 points - transitive group 30T14
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)

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

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

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

G:=TransitiveGroup(30,14);

Matrix representation of D30 in GL2(𝔽29) generated by

028
19
,
922
2820
G:=sub<GL(2,GF(29))| [0,1,28,9],[9,28,22,20] >;

D30 in GAP, Magma, Sage, TeX

D_{30}
% in TeX

G:=Group("D30");
// GroupNames label

G:=SmallGroup(60,12);
// by ID

G=gap.SmallGroup(60,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-5,98,771]);
// Polycyclic

G:=Group<a,b|a^30=b^2=1,b*a*b=a^-1>;
// generators/relations

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