Copied to
clipboard

## G = C3⋊S3⋊4D12order 432 = 24·33

### The semidirect product of C3⋊S3 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C3⋊S3⋊4D12
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — C3⋊S3⋊4D12
 Lower central C33 — C32×C6 — C3⋊S3⋊4D12
 Upper central C1 — C2

Generators and relations for C3⋊S34D12
G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2220 in 290 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×22], C6, C6 [×2], C6 [×12], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×2], C12 [×6], D6, D6 [×32], C2×C6 [×6], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×11], C3×C6, C3×C6 [×2], C3×C6 [×5], C4×S3 [×2], D12 [×8], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×7], C33, C3×Dic3 [×4], C3×Dic3 [×2], C3×C12 [×2], S32 [×14], S3×C6 [×2], S3×C6 [×10], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×9], C62, C2×D12, S3×D4 [×2], S3×C32, C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C33⋊C2, C32×C6, C6.D6, C3⋊D12 [×2], C3⋊D12 [×6], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], C12⋊S3 [×2], C2×S32 [×6], C22×C3⋊S3, C32×Dic3 [×2], S3×C3⋊S3 [×2], C324D6 [×2], S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3 [×2], C2×C33⋊C2, S3×D12 [×2], Dic3⋊D6, C3×C6.D6, C3×C3⋊D12 [×2], C338D4 [×2], C2×S3×C3⋊S3, C2×C324D6, C3⋊S34D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, D12 [×2], C22×S3 [×3], S32 [×3], C2×D12, S3×D4 [×2], C2×S32 [×3], S3×D12 [×2], Dic3⋊D6, S33, C3⋊S34D12

Permutation representations of C3⋊S34D12
On 24 points - transitive group 24T1297
Generators in S24
```(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)```

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

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

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

`G:=TransitiveGroup(24,1297);`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 6 9 9 18 18 54 2 2 2 4 4 4 8 6 6 2 2 2 4 4 4 8 12 12 12 12 18 18 36 36 6 6 6 6 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D12 S32 S32 S3×D4 C2×S32 S3×D12 Dic3⋊D6 S33 C3⋊S3⋊4D12 kernel C3⋊S3⋊4D12 C3×C6.D6 C3×C3⋊D12 C33⋊8D4 C2×S3×C3⋊S3 C2×C32⋊4D6 C6.D6 C3⋊D12 C3×C3⋊S3 C3×Dic3 S3×C6 C2×C3⋊S3 C3⋊S3 Dic3 D6 C32 C6 C3 C3 C2 C1 # reps 1 1 2 2 1 1 1 2 2 4 2 3 4 2 1 2 3 4 2 1 1

Matrix representation of C3⋊S34D12 in GL8(ℤ)

 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0
,
 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
,
 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0
,
 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0] >;`

C3⋊S34D12 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3\rtimes_4D_{12}`
`% in TeX`

`G:=Group("C3:S3:4D12");`
`// GroupNames label`

`G:=SmallGroup(432,602);`
`// by ID`

`G=gap.SmallGroup(432,602);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,254,58,298,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;`
`// generators/relations`

׿
×
𝔽