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## G = S3×D15order 180 = 22·32·5

### Direct product of S3 and D15

Aliases: S3×D15, C31D30, C152D6, C321D10, C51S32, (C5×S3)⋊S3, (C3×S3)⋊D5, C31(S3×D5), C3⋊D152C2, (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 >

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

Character table of S3×D15

 class 1 2A 2B 2C 3A 3B 3C 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 30A 30B 30C 30D size 1 3 15 45 2 2 4 2 2 6 30 6 6 2 2 2 2 4 4 4 4 4 4 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 0 -2 0 2 -1 -1 2 2 0 1 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 0 -1 2 -1 2 2 -1 0 2 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 -1 2 -1 2 2 1 0 -2 -2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 1 1 1 1 orthogonal lifted from D6 ρ8 2 0 2 0 2 -1 -1 2 2 0 -1 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ9 2 -2 0 0 2 2 2 -1+√5/2 -1-√5/2 -2 0 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 D10 ρ10 2 2 0 0 2 2 2 -1-√5/2 -1+√5/2 2 0 -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 ρ11 2 -2 0 0 2 2 2 -1-√5/2 -1+√5/2 -2 0 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 D10 ρ12 2 2 0 0 2 2 2 -1+√5/2 -1-√5/2 2 0 -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 ρ13 2 -2 0 0 -1 2 -1 -1+√5/2 -1-√5/2 1 0 1-√5/2 1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -1+√5/2 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 orthogonal lifted from D30 ρ14 2 -2 0 0 -1 2 -1 -1+√5/2 -1-√5/2 1 0 1-√5/2 1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 -1+√5/2 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 orthogonal lifted from D30 ρ15 2 2 0 0 -1 2 -1 -1+√5/2 -1-√5/2 -1 0 -1+√5/2 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 orthogonal lifted from D15 ρ16 2 2 0 0 -1 2 -1 -1+√5/2 -1-√5/2 -1 0 -1+√5/2 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 orthogonal lifted from D15 ρ17 2 -2 0 0 -1 2 -1 -1-√5/2 -1+√5/2 1 0 1+√5/2 1-√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 -1+√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 -1-√5/2 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 orthogonal lifted from D30 ρ18 2 -2 0 0 -1 2 -1 -1-√5/2 -1+√5/2 1 0 1+√5/2 1-√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 -1+√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 -1-√5/2 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 orthogonal lifted from D30 ρ19 2 2 0 0 -1 2 -1 -1-√5/2 -1+√5/2 -1 0 -1-√5/2 -1+√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 -1+√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 -1-√5/2 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal lifted from D15 ρ20 2 2 0 0 -1 2 -1 -1-√5/2 -1+√5/2 -1 0 -1-√5/2 -1+√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 -1+√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 -1-√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 orthogonal lifted from D15 ρ21 4 0 0 0 -2 -2 1 4 4 0 0 0 0 -2 -2 -2 -2 -2 1 1 1 1 -2 0 0 0 0 orthogonal lifted from S32 ρ22 4 0 0 0 4 -2 -2 -1-√5 -1+√5 0 0 0 0 -1+√5 -1-√5 -1-√5 -1+√5 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from S3×D5 ρ23 4 0 0 0 4 -2 -2 -1+√5 -1-√5 0 0 0 0 -1-√5 -1+√5 -1+√5 -1-√5 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from S3×D5 ρ24 4 0 0 0 -2 -2 1 -1+√5 -1-√5 0 0 0 0 -2ζ3ζ53+2ζ3ζ52-2ζ53 2ζ3ζ54-2ζ3ζ5-2ζ5 2ζ32ζ54-2ζ32ζ5-2ζ5 2ζ3ζ53-2ζ3ζ52-2ζ52 1+√5/2 ζ3ζ53-ζ3ζ52+ζ53 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 1-√5/2 0 0 0 0 orthogonal faithful ρ25 4 0 0 0 -2 -2 1 -1-√5 -1+√5 0 0 0 0 2ζ32ζ54-2ζ32ζ5-2ζ5 -2ζ3ζ53+2ζ3ζ52-2ζ53 2ζ3ζ53-2ζ3ζ52-2ζ52 2ζ3ζ54-2ζ3ζ5-2ζ5 1-√5/2 ζ3ζ54-ζ3ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 ζ32ζ54-ζ32ζ5+ζ54 1+√5/2 0 0 0 0 orthogonal faithful ρ26 4 0 0 0 -2 -2 1 -1+√5 -1-√5 0 0 0 0 2ζ3ζ53-2ζ3ζ52-2ζ52 2ζ32ζ54-2ζ32ζ5-2ζ5 2ζ3ζ54-2ζ3ζ5-2ζ5 -2ζ3ζ53+2ζ3ζ52-2ζ53 1+√5/2 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 1-√5/2 0 0 0 0 orthogonal faithful ρ27 4 0 0 0 -2 -2 1 -1-√5 -1+√5 0 0 0 0 2ζ3ζ54-2ζ3ζ5-2ζ5 2ζ3ζ53-2ζ3ζ52-2ζ52 -2ζ3ζ53+2ζ3ζ52-2ζ53 2ζ32ζ54-2ζ32ζ5-2ζ5 1-√5/2 ζ32ζ54-ζ32ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 ζ3ζ54-ζ3ζ5+ζ54 1+√5/2 0 0 0 0 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  D62D15  C3⋊Dic30

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

 1 0 0 0 0 1 0 0 0 0 30 1 0 0 30 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 6 22 0 0 9 28 0 0 0 0 1 0 0 0 0 1
,
 14 21 0 0 4 17 0 0 0 0 1 0 0 0 0 1
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

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