Copied to
clipboard

## G = C23⋊D12order 192 = 26·3

### The semidirect product of C23 and D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C23⋊D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C3⋊D4 — D4⋊6D6 — C23⋊D12
 Lower central C3 — C6 — C22×C6 — C23⋊D12
 Upper central C1 — C2 — C23 — C23⋊C4

Generators and relations for C23⋊D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 800 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22, C22 [×2], C22 [×18], S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], Dic3 [×3], C12 [×3], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic6, C4×S3 [×2], D12 [×5], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C23⋊C4, C23⋊C4 [×2], C22≀C2 [×3], 2+ 1+4, D6⋊C4 [×3], C6.D4, C3×C22⋊C4 [×2], C2×D12 [×2], C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4, S3×C23, C2≀C22, C23.6D6 [×2], C3×C23⋊C4, D6⋊D4 [×2], C232D6, D46D6, C23⋊D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], C2≀C22, D6⋊D4, C23⋊D12

Character table of C23⋊D12

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E size 1 1 2 2 2 4 12 12 12 12 2 4 8 8 12 12 24 2 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 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 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 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 1 1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ9 2 2 -2 2 -2 0 0 2 0 0 2 0 0 0 0 -2 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 -2 0 0 0 2 0 0 0 2 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 0 0 0 0 -1 -2 -2 2 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 2 -2 0 0 0 0 -1 -2 2 -2 0 0 0 -1 -1 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 2 0 2 0 0 0 2 0 0 0 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 2 -2 0 0 -2 0 0 2 0 0 0 0 2 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 0 0 0 0 -1 2 -2 -2 0 0 0 -1 -1 -1 -1 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ16 2 2 2 2 2 2 0 0 0 0 -1 2 2 2 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 2 -2 -2 2 0 0 0 0 2 -2 0 0 0 0 0 2 -2 2 -2 2 0 0 0 -2 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 -2 0 0 0 0 2 2 0 0 0 0 0 2 -2 2 -2 -2 0 0 0 2 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 2 0 0 0 0 -1 -2 0 0 0 0 0 -1 1 -1 1 -1 √3 √3 -√3 1 -√3 orthogonal lifted from D12 ρ20 2 2 2 -2 -2 -2 0 0 0 0 -1 2 0 0 0 0 0 -1 1 -1 1 1 √3 -√3 -√3 -1 √3 orthogonal lifted from D12 ρ21 2 2 2 -2 -2 -2 0 0 0 0 -1 2 0 0 0 0 0 -1 1 -1 1 1 -√3 √3 √3 -1 -√3 orthogonal lifted from D12 ρ22 2 2 2 -2 -2 2 0 0 0 0 -1 -2 0 0 0 0 0 -1 1 -1 1 -1 -√3 -√3 √3 1 √3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 2 -2 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ24 4 4 -4 -4 4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ26 4 -4 0 0 0 0 0 0 -2 2 4 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ27 8 -8 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23⋊D12
On 24 points - transitive group 24T336
Generators in S24
```(1 16)(2 17)(3 24)(4 13)(5 20)(6 21)(7 19)(8 14)(9 15)(10 22)(11 23)(12 18)
(1 10)(3 12)(5 8)(14 20)(16 22)(18 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 3)(4 6)(7 9)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)```

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

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

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

`G:=TransitiveGroup(24,336);`

On 24 points - transitive group 24T345
Generators in S24
```(2 16)(3 17)(6 20)(7 21)(10 24)(11 13)
(2 16)(4 18)(6 20)(8 22)(10 24)(12 14)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 19)(14 18)(15 17)(20 24)(21 23)```

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

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

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

`G:=TransitiveGroup(24,345);`

On 24 points - transitive group 24T367
Generators in S24
```(1 19)(2 23)(3 15)(4 13)(5 17)(6 21)(7 22)(8 20)(9 18)(10 16)(11 14)(12 24)
(1 10)(2 8)(3 12)(4 7)(5 11)(6 9)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)```

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

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

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

`G:=TransitiveGroup(24,367);`

Matrix representation of C23⋊D12 in GL6(𝔽13)

 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 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 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
,
 1 0 0 0 0 0 0 1 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
,
 8 7 0 0 0 0 12 9 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 4 7 0 0 0 0 9 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,12,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[4,9,0,0,0,0,7,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;`

C23⋊D12 in GAP, Magma, Sage, TeX

`C_2^3\rtimes D_{12}`
`% in TeX`

`G:=Group("C2^3:D12");`
`// GroupNames label`

`G:=SmallGroup(192,300);`
`// by ID`

`G=gap.SmallGroup(192,300);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,570,1684,438,6278]);`
`// Polycyclic`

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

Export

׿
×
𝔽