Copied to
clipboard

G = C3⋊S3.2D12order 432 = 24·33

1st non-split extension by C3⋊S3 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C3⋊S3.2D12
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C32⋊4D6 — C3⋊S3.2D12
 Lower central C33 — C3×C3⋊S3 — C3⋊S3.2D12
 Upper central C1 — C2

Generators and relations for C3⋊S3.2D12
G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=c, ab=ba, cac=ebe-1=a-1, ad=da, eae-1=b, cbc=dbd-1=b-1, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 1012 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22 [×5], S3 [×10], C6, C6 [×8], C2×C4 [×2], C23, C32, C32 [×4], Dic3 [×2], C12 [×3], D6 [×12], C2×C6 [×2], C22⋊C4, C3×S3 [×10], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3, C2×Dic3, C2×C12, C22×S3 [×2], C33, C3×Dic3 [×3], C3×C12, C32⋊C4, S32 [×7], S3×C6 [×5], C2×C3⋊S3, C2×C3⋊S3, D6⋊C4, C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, C6.D6, S3×C12, C2×C32⋊C4, C2×S32 [×2], C32×Dic3, C33⋊C4, C324D6 [×2], C324D6, C6×C3⋊S3, C6×C3⋊S3, S32⋊C4, C3×C6.D6, C2×C33⋊C4, C2×C324D6, C3⋊S3.2D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, D6⋊C4, S3≀C2, S32⋊C4, C33⋊D4, C3⋊S3.2D12

Permutation representations of C3⋊S3.2D12
On 24 points - transitive group 24T1313
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 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)
(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 22 17 6)(2 5 18 21)(3 20 19 4)(7 16 23 12)(8 11 24 15)(9 14 13 10)```

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

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

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

`G:=TransitiveGroup(24,1313);`

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 C4×S3 D12 C3⋊D4 S3≀C2 S32⋊C4 S32⋊C4 C33⋊D4 C3⋊S3.2D12 C33⋊D4 C3⋊S3.2D12 kernel C3⋊S3.2D12 C3×C6.D6 C2×C33⋊C4 C2×C32⋊4D6 C32⋊4D6 C6.D6 C3×C3⋊S3 C32×C6 C2×C3⋊S3 C3⋊S3 C3⋊S3 C3×C6 C6 C3 C3 C2 C1 C2 C1 # reps 1 1 1 1 4 1 1 1 1 2 2 2 4 2 2 4 4 1 1

Matrix representation of C3⋊S3.2D12 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 12 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 12 0 12 0 0 0 0 12 0 12
,
 1 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 12 0 12 0 0 0 0 12 0 12
,
 10 10 0 0 0 0 3 7 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0
,
 10 6 0 0 0 0 3 3 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 12 0 12 0 0 0 0 1 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,12,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,12,0,0,0,0,12,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,1,0,0,12,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,0,0,12,0] >;`

C3⋊S3.2D12 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3._2D_{12}`
`% in TeX`

`G:=Group("C3:S3.2D12");`
`// GroupNames label`

`G:=SmallGroup(432,579);`
`// by ID`

`G=gap.SmallGroup(432,579);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1684,571,298,677,1027,14118]);`
`// Polycyclic`

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

׿
×
𝔽