Copied to
clipboard

## G = C3⋊S4order 72 = 23·32

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

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 >

3C2
18C2
4C3
4C3
4C3
9C4
9C22
3C6
6S3
12S3
12S3
12S3
4C32
9D4
3D6
3Dic3
3S4
3S4
3S4

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  C324S4  (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  C324S4  (C4×C12)⋊S3  PSO4+ (𝔽3)  (C2×C6)⋊S4  A4⋊D15

Polynomial with Galois group C3⋊S4 over ℚ
actionf(x)Disc(f)
12T44x12-6x6-10x3-6-214·323·56

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

׿
×
𝔽