S3xC3^2:C4 - GroupNames

G = S₃×C₃²⋊C₄ order 216 = 2³·3³

Direct product of S₃ and C₃²⋊C₄

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

Aliases: S₃×C₃²⋊C₄, C₃³⋊(C₂×C₄), (S₃×C₃²)⋊C₄, C₃³⋊C₂⋊C₄, C₃⋊S₃.₄D₆, C₃²⋊₅(C₄×S₃), C₃³⋊C₄⋊₁C₂, (S₃×C₃⋊S₃).C₂, C₃⋊₁(C₂×C₃²⋊C₄), (C₃×C₃²⋊C₄)⋊₂C₂, (C₃×C₃⋊S₃).₃C₂², SmallGroup(216,156)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×C₃²⋊C₄

Chief series

C₁ — C₃ — C₃³ — C₃×C₃⋊S₃ — S₃×C₃⋊S₃ — S₃×C₃²⋊C₄

Lower central

C₃³ — S₃×C₃²⋊C₄

Upper central

C₁

Generators and relations for S₃×C₃²⋊C₄
G = < a,b,c,d,e | a³=b²=c³=d³=e⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede^-1=cd=dc, ece^-1=c^-1d >

Subgroups: 468 in 60 conjugacy classes, 14 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², S₃, S₃, C₆, C₂×C₄, C₃², C₃², Dic₃, C₁₂, D₆, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₄×S₃, C₃³, C₃²⋊C₄, C₃²⋊C₄, S₃², C₂×C₃⋊S₃, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₂×C₃²⋊C₄, C₃×C₃²⋊C₄, C₃³⋊C₄, S₃×C₃⋊S₃, S₃×C₃²⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₆, C₄×S₃, C₃²⋊C₄, C₂×C₃²⋊C₄, S₃×C₃²⋊C₄

Character table of S₃×C₃²⋊C₄

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	12A	12B
size	1	3	9	27	2	4	4	8	8	9	9	27	27	12	12	18	18	18
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	1	1	-1	-1	1	1	1	1	1	i	-i	-i	i	1	1	-1	-i	i	linear of order 4
ρ₆	1	-1	-1	1	1	1	1	1	1	i	-i	i	-i	-1	-1	-1	-i	i	linear of order 4
ρ₇	1	-1	-1	1	1	1	1	1	1	-i	i	-i	i	-1	-1	-1	i	-i	linear of order 4
ρ₈	1	1	-1	-1	1	1	1	1	1	-i	i	i	-i	1	1	-1	i	-i	linear of order 4
ρ₉	2	0	2	0	-1	2	2	-1	-1	2	2	0	0	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₁₀	2	0	2	0	-1	2	2	-1	-1	-2	-2	0	0	0	0	-1	1	1	orthogonal lifted from D₆
ρ₁₁	2	0	-2	0	-1	2	2	-1	-1	-2i	2i	0	0	0	0	1	-i	i	complex lifted from C₄×S₃
ρ₁₂	2	0	-2	0	-1	2	2	-1	-1	2i	-2i	0	0	0	0	1	i	-i	complex lifted from C₄×S₃
ρ₁₃	4	4	0	0	4	-2	1	1	-2	0	0	0	0	1	-2	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₁₄	4	-4	0	0	4	1	-2	-2	1	0	0	0	0	2	-1	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₅	4	-4	0	0	4	-2	1	1	-2	0	0	0	0	-1	2	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₆	4	4	0	0	4	1	-2	-2	1	0	0	0	0	-2	1	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₁₇	8	0	0	0	-4	-4	2	-1	2	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₈	8	0	0	0	-4	2	-4	2	-1	0	0	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of S₃×C₃²⋊C₄
►On 12 points - transitive group 12T119

Generators in S₁₂

(1 8 9)(2 5 10)(3 6 11)(4 7 12)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)

(1 8 9)(3 11 6)

(1 8 9)(2 10 5)(3 11 6)(4 7 12)

(1 2 3 4)(5 6 7 8)(9 10 11 12)

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

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

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

G:=TransitiveGroup(12,119);

►On 18 points - transitive group 18T95

Generators in S₁₈

(1 6 3)(2 5 4)(7 11 18)(8 12 15)(9 13 16)(10 14 17)

(1 6)(2 5)(7 18)(8 15)(9 16)(10 17)

(1 16 18)(2 17 15)(3 13 11)(4 14 12)(5 10 8)(6 9 7)

(1 18 16)(3 11 13)(6 7 9)

(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

G:=sub<Sym(18)| (1,6,3)(2,5,4)(7,11,18)(8,12,15)(9,13,16)(10,14,17), (1,6)(2,5)(7,18)(8,15)(9,16)(10,17), (1,16,18)(2,17,15)(3,13,11)(4,14,12)(5,10,8)(6,9,7), (1,18,16)(3,11,13)(6,7,9), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;

G:=Group( (1,6,3)(2,5,4)(7,11,18)(8,12,15)(9,13,16)(10,14,17), (1,6)(2,5)(7,18)(8,15)(9,16)(10,17), (1,16,18)(2,17,15)(3,13,11)(4,14,12)(5,10,8)(6,9,7), (1,18,16)(3,11,13)(6,7,9), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );

G=PermutationGroup([[(1,6,3),(2,5,4),(7,11,18),(8,12,15),(9,13,16),(10,14,17)], [(1,6),(2,5),(7,18),(8,15),(9,16),(10,17)], [(1,16,18),(2,17,15),(3,13,11),(4,14,12),(5,10,8),(6,9,7)], [(1,18,16),(3,11,13),(6,7,9)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])

G:=TransitiveGroup(18,95);

►On 24 points - transitive group 24T559

Generators in S₂₄

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

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

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

(2 18 13)(4 15 20)(5 23 10)(7 12 21)

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

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

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

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

G:=TransitiveGroup(24,559);

►On 27 points - transitive group 27T85

Generators in S₂₇

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

(2 3)(4 19)(5 16)(6 17)(7 18)(8 20)(9 21)(10 22)(11 23)

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

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

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

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

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

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

G:=TransitiveGroup(27,85);

S₃×C₃²⋊C₄ is a maximal quotient of D₆⋊(C₃²⋊C₄) C₃³⋊(C₄⋊C₄) C₃³⋊₅(C₂×C₈) C₃³⋊M₄(2) C₃³⋊₂M₄(2)

Polynomial with Galois group S₃×C₃²⋊C₄ over ℚ

action	f(x)	Disc(f)
12T119	x¹²-x⁹-4x⁶+4x³+1	3¹⁸·5⁹

Matrix representation of S₃×C₃²⋊C₄ ►in GL₆(𝔽₁₃)

12	1	0	0	0	0
12	0	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	1

0	12	0	0	0	0
12	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	1
0	0	12	1	12	1
0	0	0	12	0	0
0	0	1	12	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	1	0	0
0	0	12	0	0	0
0	0	0	0	12	1
0	0	0	0	12	0

5	0	0	0	0	0
0	5	0	0	0	0
0	0	0	8	0	0
0	0	0	0	0	5
0	0	5	0	0	0
0	0	0	0	8	0

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,1,0,0,1,1,12,12,0,0,0,12,0,0,0,0,1,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,5,0,0] >;

S₃×C₃²⋊C₄ in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes C_4

% in TeX

G:=Group("S3xC3^2:C4");

// GroupNames label

G:=SmallGroup(216,156);

// by ID

G=gap.SmallGroup(216,156);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,31,489,111,490,376,5189]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;

// generators/relations

Export

Character table of S₃×C₃²⋊C₄ in TeX

G = S3×C32⋊C4 order 216 = 23·33

Direct product of S3 and C32⋊C4

G = S₃×C₃²⋊C₄ order 216 = 2³·3³

Direct product of S₃ and C₃²⋊C₄