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## G = C24⋊1D6order 288 = 25·32

### 1st semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊1D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — C24⋊1D6
 Lower central C32 — C3×C6 — C3×C12 — C24⋊1D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C241D6
G = < a,b,c | a24=b6=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >

Subgroups: 834 in 147 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×8], C6 [×2], C6 [×3], C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×11], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, C4×S3 [×2], D12, D12 [×8], C3⋊D4 [×2], C2×C12, C3×D4, C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3 [×2], C8⋊S3, C24⋊C2, C24⋊C2, D24 [×4], D4⋊S3, Q82S3, C3×M4(2), C3×SD16, C2×D12, C4○D12, S3×D4, Q83S3, C3×C3⋊C8, C3×C24, C6.D6, C3⋊D12 [×2], C3×Dic6, S3×C12, C3×D12, C12⋊S3 [×2], C2×S32, C8⋊D6, Q83D6, C3⋊D24, C325SD16, C3×C8⋊S3, C3×C24⋊C2, C325D8, D6.6D6, S3×D12, C241D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, S3×D4, C2×S32, C8⋊D6, Q83D6, S3×D12, C241D6

Smallest permutation representation of C241D6
On 48 points
Generators in S48
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 38 17 46 9 30)(2 25 18 33 10 41)(3 36 19 44 11 28)(4 47 20 31 12 39)(5 34 21 42 13 26)(6 45 22 29 14 37)(7 32 23 40 15 48)(8 43 24 27 16 35)
(1 8)(2 7)(3 6)(4 5)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)```

`G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,38,17,46,9,30)(2,25,18,33,10,41)(3,36,19,44,11,28)(4,47,20,31,12,39)(5,34,21,42,13,26)(6,45,22,29,14,37)(7,32,23,40,15,48)(8,43,24,27,16,35), (1,8)(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;`

`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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,38,17,46,9,30)(2,25,18,33,10,41)(3,36,19,44,11,28)(4,47,20,31,12,39)(5,34,21,42,13,26)(6,45,22,29,14,37)(7,32,23,40,15,48)(8,43,24,27,16,35), (1,8)(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );`

`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),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,38,17,46,9,30),(2,25,18,33,10,41),(3,36,19,44,11,28),(4,47,20,31,12,39),(5,34,21,42,13,26),(6,45,22,29,14,37),(7,32,23,40,15,48),(8,43,24,27,16,35)], [(1,8),(2,7),(3,6),(4,5),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)])`

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A ··· 24H 24I 24J order 1 2 2 2 2 2 3 3 3 4 4 4 6 6 6 6 6 8 8 12 12 12 12 12 12 12 24 ··· 24 24 24 size 1 1 6 12 36 36 2 2 4 2 6 12 2 2 4 12 24 4 12 2 2 4 4 4 12 24 4 ··· 4 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 D12 D12 C8⋊C22 S32 S3×D4 C2×S32 C8⋊D6 Q8⋊3D6 S3×D12 C24⋊1D6 kernel C24⋊1D6 C3⋊D24 C32⋊5SD16 C3×C8⋊S3 C3×C24⋊C2 C32⋊5D8 D6.6D6 S3×D12 C8⋊S3 C24⋊C2 C3×Dic3 S3×C6 C3⋊C8 C24 Dic6 C4×S3 D12 Dic3 D6 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 1 2 2 2 4

Matrix representation of C241D6 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 59 66 0 0 0 0 0 0 7 66 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0
,
 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 7 66 0 0 0 0 0 0 0 0 50 23 0 53 0 0 0 0 23 23 53 0 0 0 0 0 0 20 50 50 0 0 0 0 20 0 50 23
,
 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 7 66 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0

`G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,59,7,0,0,0,0,0,0,66,66,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0,0,0,0,0,0,0,50,23,0,20,0,0,0,0,23,23,20,0,0,0,0,0,0,53,50,50,0,0,0,0,53,0,50,23],[0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0] >;`

C241D6 in GAP, Magma, Sage, TeX

`C_{24}\rtimes_1D_6`
`% in TeX`

`G:=Group("C24:1D6");`
`// GroupNames label`

`G:=SmallGroup(288,442);`
`// by ID`

`G=gap.SmallGroup(288,442);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,58,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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