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## G = C24⋊3D6order 288 = 25·32

### 3rd semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊3D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C24⋊3D6
 Lower central C32 — C3×C6 — C3×C12 — C24⋊3D6
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C243D6
G = < a,b,c | a24=b6=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >

Subgroups: 1100 in 204 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×2], C4, C22, C22 [×5], S3 [×12], C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3 [×4], C12 [×8], D6 [×20], C2×C6 [×4], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊S3 [×3], C3×C6, C3×C6, C24 [×8], Dic6 [×4], C4×S3 [×4], D12 [×16], C3⋊D4 [×4], C2×C12 [×4], C22×S3 [×4], C8⋊C22, C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×5], C62, C24⋊C2 [×8], D24 [×8], C3×M4(2) [×4], C2×D12 [×4], C4○D12 [×4], C3×C24 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C12⋊S3 [×2], C12⋊S3, C327D4, C6×C12, C22×C3⋊S3, C8⋊D6 [×4], C242S3 [×2], C325D8 [×2], C32×M4(2), C2×C12⋊S3, C12.59D6, C243D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C8⋊D6 [×4], C2×C12⋊S3, C243D6

Smallest permutation representation of C243D6
On 72 points
Generators in S72
```(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 25 67 13 37 55)(2 38 68)(3 27 69 15 39 57)(4 40 70)(5 29 71 17 41 59)(6 42 72)(7 31 49 19 43 61)(8 44 50)(9 33 51 21 45 63)(10 46 52)(11 35 53 23 47 65)(12 48 54)(14 26 56)(16 28 58)(18 30 60)(20 32 62)(22 34 64)(24 36 66)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(38 48)(39 47)(40 46)(41 45)(42 44)```

`G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,25,67,13,37,55)(2,38,68)(3,27,69,15,39,57)(4,40,70)(5,29,71,17,41,59)(6,42,72)(7,31,49,19,43,61)(8,44,50)(9,33,51,21,45,63)(10,46,52)(11,35,53,23,47,65)(12,48,54)(14,26,56)(16,28,58)(18,30,60)(20,32,62)(22,34,64)(24,36,66), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(38,48)(39,47)(40,46)(41,45)(42,44)>;`

`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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,25,67,13,37,55)(2,38,68)(3,27,69,15,39,57)(4,40,70)(5,29,71,17,41,59)(6,42,72)(7,31,49,19,43,61)(8,44,50)(9,33,51,21,45,63)(10,46,52)(11,35,53,23,47,65)(12,48,54)(14,26,56)(16,28,58)(18,30,60)(20,32,62)(22,34,64)(24,36,66), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(38,48)(39,47)(40,46)(41,45)(42,44) );`

`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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,25,67,13,37,55),(2,38,68),(3,27,69,15,39,57),(4,40,70),(5,29,71,17,41,59),(6,42,72),(7,31,49,19,43,61),(8,44,50),(9,33,51,21,45,63),(10,46,52),(11,35,53,23,47,65),(12,48,54),(14,26,56),(16,28,58),(18,30,60),(20,32,62),(22,34,64),(24,36,66)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(38,48),(39,47),(40,46),(41,45),(42,44)])`

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D12 D12 C8⋊C22 C8⋊D6 kernel C24⋊3D6 C24⋊2S3 C32⋊5D8 C32×M4(2) C2×C12⋊S3 C12.59D6 C3×M4(2) C3×C12 C62 C24 C2×C12 C12 C2×C6 C32 C3 # reps 1 2 2 1 1 1 4 1 1 8 4 8 8 1 8

Matrix representation of C243D6 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 7 59 0 0 0 0 14 66 0 0
,
 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 72 1 0 0 0 0 0 1 0 0 0 0 0 0 66 66 0 0 0 0 59 7

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,66,59,0,0,0,0,66,7] >;`

C243D6 in GAP, Magma, Sage, TeX

`C_{24}\rtimes_3D_6`
`% in TeX`

`G:=Group("C24:3D6");`
`// GroupNames label`

`G:=SmallGroup(288,765);`
`// by ID`

`G=gap.SmallGroup(288,765);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,675,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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