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G = D15⋊S3  order 180 = 22·32·5

The semidirect product of D15 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: D15⋊S3, C15⋊3D6, C32⋊2D10, C5⋊2S32, C3⋊S3⋊D5, C3⋊3(S3×D5), (C3×D15)⋊3C2, (C3×C15)⋊4C22, (C5×C3⋊S3)⋊2C2, SmallGroup(180,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C15 — D15⋊S3
Chief series
C1 — C5 — C15 — C3×C15 — C3×D15 — D15⋊S3
Lower central
C3×C15 — D15⋊S3
Upper central
C1

Generators and relations for D15⋊S3
 G = < a,b,c,d | a15=b2=c3=d2=1, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c-1 >

C1
9C2
15C2
15C2
C3
C3
2C3
C5
45C22
3S3
3S3
5S3
5S3
6S3
15C6
15C6
C32
3D5
3D5
9C10
C15
C15
2C15
15D6
15D6
C3⋊S3
5C3×S3
5C3×S3
9D10
D15
D15
3C3×D5
3C5×S3
3C5×S3
3C3×D5
6C5×S3
C3×C15
5S32
3S3×D5
3S3×D5
C3×D15
C3×D15
C5×C3⋊S3
D15⋊S3

Character table of D15⋊S3

 class 12A2B2C3A3B3C5A5B6A6B10A10B15A15B15C15D15E15F15G15H
 size 191515224223030181844444444
ρ1111111111111111111111    trivial
ρ21-11-1111111-1-1-111111111    linear of order 2
ρ311-1-111111-1-11111111111    linear of order 2
ρ41-1-1111111-11-1-111111111    linear of order 2
ρ5200-22-1-1220100-1-1-1-122-1-1    orthogonal lifted from D6
ρ62020-12-122-1000-12-1-1-1-12-1    orthogonal lifted from S3
ρ720022-1-1220-100-1-1-1-122-1-1    orthogonal lifted from S3
ρ820-20-12-1221000-12-1-1-1-12-1    orthogonal lifted from D6
ρ92200222-1-√5/2-1+√5/200-1-√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
ρ102-200222-1+√5/2-1-√5/2001-√5/21+√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 D10
ρ112-200222-1-√5/2-1+√5/2001+√5/21-√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 D10
ρ122200222-1+√5/2-1-√5/200-1+√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
ρ134000-2-214400001-211-2-2-21    orthogonal lifted from S32
ρ144000-24-2-1-√5-1+√500001+√5/2-1+√51+√5/21-√5/21-√5/21+√5/2-1-√51-√5/2    orthogonal lifted from S3×D5
ρ1540004-2-2-1+√5-1-√500001-√5/21+√5/21-√5/21+√5/2-1-√5-1+√51-√5/21+√5/2    orthogonal lifted from S3×D5
ρ1640004-2-2-1-√5-1+√500001+√5/21-√5/21+√5/21-√5/2-1+√5-1-√51+√5/21-√5/2    orthogonal lifted from S3×D5
ρ174000-24-2-1+√5-1-√500001-√5/2-1-√51-√5/21+√5/21+√5/21-√5/2-1+√51+√5/2    orthogonal lifted from S3×D5
ρ184000-2-21-1+√5-1-√500002ζ54-ζ51+√5/2-ζ54+2ζ52ζ53-ζ521+√5/21-√5/21-√5/2-ζ53+2ζ52    complex faithful
ρ194000-2-21-1-√5-1+√500002ζ53-ζ521-√5/2-ζ53+2ζ52-ζ54+2ζ51-√5/21+√5/21+√5/22ζ54-ζ5    complex faithful
ρ204000-2-21-1+√5-1-√50000-ζ54+2ζ51+√5/22ζ54-ζ5-ζ53+2ζ521+√5/21-√5/21-√5/22ζ53-ζ52    complex faithful
ρ214000-2-21-1-√5-1+√50000-ζ53+2ζ521-√5/22ζ53-ζ522ζ54-ζ51-√5/21+√5/21+√5/2-ζ54+2ζ5    complex faithful

Permutation representations of D15⋊S3
►On 30 points - transitive group 30T43
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)

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

(2 12)(3 8)(5 15)(6 11)(9 14)(17 27)(18 23)(20 30)(21 26)(24 29)
 
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (2,12)(3,8)(5,15)(6,11)(9,14)(17,27)(18,23)(20,30)(21,26)(24,29)>;
 
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (2,12)(3,8)(5,15)(6,11)(9,14)(17,27)(18,23)(20,30)(21,26)(24,29) );
 
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16)], [(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(2,12),(3,8),(5,15),(6,11),(9,14),(17,27),(18,23),(20,30),(21,26),(24,29)]])
 
G:=TransitiveGroup(30,43);
 

D15⋊S3 is a maximal subgroup of   S32⋊D5  C32⋊D20  S32×D5
D15⋊S3 is a maximal quotient of   D30.S3  Dic15⋊S3  D30⋊S3  C32⋊3D20  C32⋊3Dic10

Matrix representation of D15⋊S3 ►in GL6(𝔽31)

12120000
2460000
0003000
0013000
000010
000001
,
6190000
21250000
0030100
000100
000010
000001
,
100000
010000
001000
000100
000001
00003030
,
3000000
0300000
000100
001000
000010
00003030

G:=sub<GL(6,GF(31))| [12,24,0,0,0,0,12,6,0,0,0,0,0,0,0,1,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,21,0,0,0,0,19,25,0,0,0,0,0,0,30,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,1,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,30,0,0,0,0,0,30] >;
 

D15⋊S3 in GAP, Magma, Sage, TeX 

D_{15}\rtimes S_3
 
% in TeX
 
G:=Group("D15:S3");
 
// GroupNames label
 
G:=SmallGroup(180,30);
 
// by ID
 
G=gap.SmallGroup(180,30);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-5,122,67,248,3604]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^15=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D15⋊S3 in TeX
Character table of D15⋊S3 in TeX

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