C3:S4 - GroupNames

G = C₃⋊S₄ order 72 = 2³·3²

The semidirect product of C₃ and S₄ acting via S₄/A₄=C₂

non-abelian, soluble, monomial, rational

Aliases: C₃⋊S₄, A₄⋊S₃, (C₂×C₆)⋊₂S₃, (C₃×A₄)⋊₂C₂, C₂²⋊(C₃⋊S₃), SmallGroup(72,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×A₄ — C₃⋊S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃×A₄ — C₃⋊S₄

Lower central

C₃×A₄ — C₃⋊S₄

Upper central

C₁

Generators and relations for C₃⋊S₄
G = < a,b,c,d,e | a³=b²=c²=d³=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, dbd^-1=ebe=bc=cb, dcd^-1=b, ce=ec, ede=d^-1 >

C₁

₃C₂

₁₈C₂

C₃

₄C₃

C₂²

₉C₄

₉C₂²

₃C₆

₆S₃

₁₂S₃

₄C₃²

₉D₄

A₄

C₂×C₆

A₄

₃D₆

₃Dic₃

₄C₃⋊S₃

₃S₄

₃C₃⋊D₄

₃S₄

C₃×A₄

C₃⋊S₄

Character table of C₃⋊S₄

class	1	2A	2B	3A	3B	3C	3D	4	6
size	1	3	18	2	8	8	8	18	6
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	1	linear of order 2
ρ₃	2	2	0	2	-1	-1	-1	0	2	orthogonal lifted from S₃
ρ₄	2	2	0	-1	2	-1	-1	0	-1	orthogonal lifted from S₃
ρ₅	2	2	0	-1	-1	-1	2	0	-1	orthogonal lifted from S₃
ρ₆	2	2	0	-1	-1	2	-1	0	-1	orthogonal lifted from S₃
ρ₇	3	-1	-1	3	0	0	0	1	-1	orthogonal lifted from S₄
ρ₈	3	-1	1	3	0	0	0	-1	-1	orthogonal lifted from S₄
ρ₉	6	-2	0	-3	0	0	0	0	1	orthogonal faithful

Permutation representations of C₃⋊S₄
►On 12 points - transitive group 12T44

Generators in S₁₂

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

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

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

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

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

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

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

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

G:=TransitiveGroup(12,44);

►On 18 points - transitive group 18T37

Generators in S₁₈

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

(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)

(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)

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

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

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

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

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

G:=TransitiveGroup(18,37);

►On 18 points - transitive group 18T40

Generators in S₁₈

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

(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)

(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)

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

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

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

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

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

G:=TransitiveGroup(18,40);

►On 24 points - transitive group 24T79

Generators in S₂₄

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,79);

C₃⋊S₄ is a maximal subgroup of
S₃×S₄ C₆²⋊S₃ C₉⋊S₄ C₃²⋊₄S₄ (C₄×C₁₂)⋊S₃ PSO₄⁺ (𝔽₃) (C₂×C₆)⋊S₄ ΓL₂(𝔽₄) A₄⋊D₁₅
C₃⋊S₄ is a maximal quotient of
C₆.₅S₄ C₆.₆S₄ C₆.₇S₄ C₉⋊S₄ C₃².₃S₄ C₃²⋊S₄ C₃²⋊₄S₄ (C₄×C₁₂)⋊S₃ PSO₄⁺ (𝔽₃) (C₂×C₆)⋊S₄ A₄⋊D₁₅

Polynomial with Galois group C₃⋊S₄ over ℚ

action	f(x)	Disc(f)
12T44	x¹²-6x⁶-10x³-6	-2¹⁴·3²³·5⁶

Matrix representation of C₃⋊S₄ ►in GL₅(ℤ)

-1	1	0	0	0
-1	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	-1	-1	-1
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	-1	-1	-1
0	0	0	0	1
0	0	0	1	0

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	-1	-1	-1
0	0	0	1	0

-1	0	0	0	0
-1	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,Integers())| [-1,-1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,-1,0,0,0,1,-1,0],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,-1,0,1,0,0,-1,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,0,-1,1,0,0,0,-1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₃⋊S₄ in GAP, Magma, Sage, TeX

C_3\rtimes S_4

% in TeX

G:=Group("C3:S4");

// GroupNames label

G:=SmallGroup(72,43);

// by ID

G=gap.SmallGroup(72,43);

# by ID

G:=PCGroup([5,-2,-3,-3,-2,2,41,182,723,368,454,684]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃⋊S₄ in TeX
Character table of C₃⋊S₄ in TeX

G = C3⋊S4 order 72 = 23·32

The semidirect product of C3 and S4 acting via S4/A4=C2

G = C₃⋊S₄ order 72 = 2³·3²

The semidirect product of C₃ and S₄ acting via S₄/A₄=C₂