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G = S3×D15  order 180 = 22·32·5

Direct product of S3 and D15

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×D15, C3⋊1D30, C15⋊2D6, C32⋊1D10, C5⋊1S32, (C5×S3)⋊S3, (C3×S3)⋊D5, C3⋊1(S3×D5), C3⋊D15⋊2C2, (S3×C15)⋊1C2, (C3×D15)⋊2C2, (C3×C15)⋊3C22, SmallGroup(180,29)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for S3×D15
 G = < a,b,c,d | a3=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
3C2
15C2
45C2
C3
C3
2C3
C5
45C22
S3
3C6
5S3
15S3
15C6
15S3
30S3
C32
3C10
3D5
9D5
C15
C15
2C15
15D6
15D6
C3×S3
5C3⋊S3
5C3×S3
9D10
D15
C5×S3
3D15
3C3×D5
3D15
3C30
6D15
C3×C15
5S32
3D30
3S3×D5
C3×D15
C3⋊D15
S3×C15
S3×D15

Character table of S3×D15

 class 12A2B2C3A3B3C5A5B6A6B10A10B15A15B15C15D15E15F15G15H15I15J30A30B30C30D
 size 131545224226306622224444446666
ρ1111111111111111111111111111    trivial
ρ211-1-1111111-11111111111111111    linear of order 2
ρ31-1-1111111-1-1-1-11111111111-1-1-1-1    linear of order 2
ρ41-11-111111-11-1-11111111111-1-1-1-1    linear of order 2
ρ520-202-1-12201002222-1-1-1-1-1-10000    orthogonal lifted from D6
ρ62200-12-122-1022-1-1-1-12-1-1-1-12-1-1-1-1    orthogonal lifted from S3
ρ72-200-12-12210-2-2-1-1-1-12-1-1-1-121111    orthogonal lifted from D6
ρ820202-1-1220-1002222-1-1-1-1-1-10000    orthogonal lifted from S3
ρ92-200222-1+√5/2-1-√5/2-201-√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+√5/2-1-√5/2-1+√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ102200222-1-√5/2-1+√5/220-1-√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-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ112-200222-1-√5/2-1+√5/2-201+√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-√5/2-1+√5/2-1-√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ122200222-1+√5/2-1-√5/220-1+√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+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ132-200-12-1-1+√5/2-1-√5/2101-√5/21+√5/2-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-1-√5/2-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-1+√5/2-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ53-ζ3ζ52+ζ53ζ32ζ54-ζ32ζ5+ζ54ζ3ζ54-ζ3ζ5+ζ54    orthogonal lifted from D30
ρ142-200-12-1-1+√5/2-1-√5/2101-√5/21+√5/2ζ3ζ53-ζ3ζ52-ζ52ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53-1-√5/2ζ3ζ53-ζ3ζ52-ζ52ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53-1+√5/2ζ3ζ53-ζ3ζ52+ζ53-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ54-ζ3ζ5+ζ54ζ32ζ54-ζ32ζ5+ζ54    orthogonal lifted from D30
ρ152200-12-1-1+√5/2-1-√5/2-10-1+√5/2-1-√5/2-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-1-√5/2-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-1+√5/2ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5    orthogonal lifted from D15
ρ162200-12-1-1+√5/2-1-√5/2-10-1+√5/2-1-√5/2ζ3ζ53-ζ3ζ52-ζ52ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53-1-√5/2ζ3ζ53-ζ3ζ52-ζ52ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53-1+√5/2-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5    orthogonal lifted from D15
ρ172-200-12-1-1-√5/2-1+√5/2101+√5/21-√5/2ζ3ζ54-ζ3ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53ζ32ζ54-ζ32ζ5-ζ5-1+√5/2ζ3ζ54-ζ3ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53ζ32ζ54-ζ32ζ5-ζ5-1-√5/2ζ3ζ54-ζ3ζ5+ζ54ζ32ζ54-ζ32ζ5+ζ54-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ53-ζ3ζ52+ζ53    orthogonal lifted from D30
ρ182-200-12-1-1-√5/2-1+√5/2101+√5/21-√5/2ζ32ζ54-ζ32ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52ζ3ζ54-ζ3ζ5-ζ5-1+√5/2ζ32ζ54-ζ32ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52ζ3ζ54-ζ3ζ5-ζ5-1-√5/2ζ32ζ54-ζ32ζ5+ζ54ζ3ζ54-ζ3ζ5+ζ54ζ3ζ53-ζ3ζ52+ζ53-ζ3ζ53+ζ3ζ52+ζ52    orthogonal lifted from D30
ρ192200-12-1-1-√5/2-1+√5/2-10-1-√5/2-1+√5/2ζ3ζ54-ζ3ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53ζ32ζ54-ζ32ζ5-ζ5-1+√5/2ζ3ζ54-ζ3ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53ζ32ζ54-ζ32ζ5-ζ5-1-√5/2ζ32ζ54-ζ32ζ5-ζ5ζ3ζ54-ζ3ζ5-ζ5ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53    orthogonal lifted from D15
ρ202200-12-1-1-√5/2-1+√5/2-10-1-√5/2-1+√5/2ζ32ζ54-ζ32ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52ζ3ζ54-ζ3ζ5-ζ5-1+√5/2ζ32ζ54-ζ32ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52ζ3ζ54-ζ3ζ5-ζ5-1-√5/2ζ3ζ54-ζ3ζ5-ζ5ζ32ζ54-ζ32ζ5-ζ5-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52    orthogonal lifted from D15
ρ214000-2-21440000-2-2-2-2-21111-20000    orthogonal lifted from S32
ρ2240004-2-2-1-√5-1+√50000-1+√5-1-√5-1-√5-1+√51-√5/21-√5/21+√5/21+√5/21-√5/21+√5/20000    orthogonal lifted from S3×D5
ρ2340004-2-2-1+√5-1-√50000-1-√5-1+√5-1+√5-1-√51+√5/21+√5/21-√5/21-√5/21+√5/21-√5/20000    orthogonal lifted from S3×D5
ρ244000-2-21-1+√5-1-√50000-2ζ3ζ53+2ζ3ζ52-2ζ532ζ3ζ54-2ζ3ζ5-2ζ52ζ32ζ54-2ζ32ζ5-2ζ52ζ3ζ53-2ζ3ζ52-2ζ521+√5/2ζ3ζ53-ζ3ζ52+ζ53ζ32ζ54-ζ32ζ5+ζ54ζ3ζ54-ζ3ζ5+ζ54-ζ3ζ53+ζ3ζ52+ζ521-√5/20000    orthogonal faithful
ρ254000-2-21-1-√5-1+√500002ζ32ζ54-2ζ32ζ5-2ζ5-2ζ3ζ53+2ζ3ζ52-2ζ532ζ3ζ53-2ζ3ζ52-2ζ522ζ3ζ54-2ζ3ζ5-2ζ51-√5/2ζ3ζ54-ζ3ζ5+ζ54ζ3ζ53-ζ3ζ52+ζ53-ζ3ζ53+ζ3ζ52+ζ52ζ32ζ54-ζ32ζ5+ζ541+√5/20000    orthogonal faithful
ρ264000-2-21-1+√5-1-√500002ζ3ζ53-2ζ3ζ52-2ζ522ζ32ζ54-2ζ32ζ5-2ζ52ζ3ζ54-2ζ3ζ5-2ζ5-2ζ3ζ53+2ζ3ζ52-2ζ531+√5/2-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ54-ζ3ζ5+ζ54ζ32ζ54-ζ32ζ5+ζ54ζ3ζ53-ζ3ζ52+ζ531-√5/20000    orthogonal faithful
ρ274000-2-21-1-√5-1+√500002ζ3ζ54-2ζ3ζ5-2ζ52ζ3ζ53-2ζ3ζ52-2ζ52-2ζ3ζ53+2ζ3ζ52-2ζ532ζ32ζ54-2ζ32ζ5-2ζ51-√5/2ζ32ζ54-ζ32ζ5+ζ54-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ53-ζ3ζ52+ζ53ζ3ζ54-ζ3ζ5+ζ541+√5/20000    orthogonal faithful

Permutation representations of S3×D15
►On 30 points - transitive group 30T42
Generators in S30
(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)

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

(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)
 
G:=sub<Sym(30)| (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), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)>;
 
G:=Group( (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), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24) );
 
G=PermutationGroup([[(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)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24)]])
 
G:=TransitiveGroup(30,42);
 

S3×D15 is a maximal subgroup of   S32×D5
S3×D15 is a maximal quotient of   C6.D30  D6⋊D15  C3⋊D60  D6⋊2D15  C3⋊Dic30

Matrix representation of S3×D15 ►in GL4(𝔽31) generated by

1000
0100
00301
00300
,
1000
0100
0001
0010
,
62200
92800
0010
0001
,
142100
41700
0010
0001
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[6,9,0,0,22,28,0,0,0,0,1,0,0,0,0,1],[14,4,0,0,21,17,0,0,0,0,1,0,0,0,0,1] >;
 

S3×D15 in GAP, Magma, Sage, TeX 

S_3\times D_{15}
 
% in TeX
 
G:=Group("S3xD15");
 
// GroupNames label
 
G:=SmallGroup(180,29);
 
// by ID
 
G=gap.SmallGroup(180,29);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-5,67,483,3604]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

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

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