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## G = C24⋊7D6order 288 = 25·32

### 7th semidirect product of C24 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C24⋊7D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — D4×C3⋊S3 — C24⋊7D6
 Lower central C32 — C3×C6 — C3×C12 — C24⋊7D6
 Upper central C1 — C2 — C4 — SD16

Generators and relations for C247D6
G = < a,b,c | a24=b6=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

Subgroups: 1036 in 204 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×4], C3 [×4], C4, C4 [×2], C22 [×6], S3 [×12], C6 [×4], C6 [×4], C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3 [×4], C12 [×4], C12 [×4], D6 [×20], C2×C6 [×4], M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3⋊S3 [×3], C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×8], D12 [×12], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C22×S3 [×4], C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×4], C62, C8⋊S3 [×4], D24 [×4], D4⋊S3 [×4], Q82S3 [×4], C3×SD16 [×4], S3×D4 [×4], Q83S3 [×4], C324C8, C3×C24, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3 [×2], C12⋊S3, C327D4, D4×C32, Q8×C32, C22×C3⋊S3, Q83D6 [×4], C24⋊S3, C325D8, C327D8, C3211SD16, C32×SD16, D4×C3⋊S3, C12.26D6, C247D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, Q83D6 [×4], D4×C3⋊S3, C247D6

Smallest permutation representation of C247D6
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 60 39)(2 55 40 20 61 34)(3 50 41 15 62 29)(4 69 42 10 63 48)(5 64 43)(6 59 44 24 65 38)(7 54 45 19 66 33)(8 49 46 14 67 28)(9 68 47)(11 58 25 23 70 37)(12 53 26 18 71 32)(13 72 27)(16 57 30 22 51 36)(17 52 31)(21 56 35)
(1 27)(2 26)(3 25)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(49 59)(50 58)(51 57)(52 56)(53 55)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)```

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

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

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

39 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 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H 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 12 12 24 ··· 24 size 1 1 4 18 36 36 2 2 2 2 2 4 18 2 2 2 2 8 8 8 8 4 36 4 4 4 4 8 8 8 8 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C8⋊C22 S3×D4 Q8⋊3D6 kernel C24⋊7D6 C24⋊S3 C32⋊5D8 C32⋊7D8 C32⋊11SD16 C32×SD16 D4×C3⋊S3 C12.26D6 C3×SD16 C3⋊Dic3 C2×C3⋊S3 C24 C3×D4 C3×Q8 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 1 4 4 4 1 4 8

Matrix representation of C247D6 in GL8(ℤ)

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

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

C247D6 in GAP, Magma, Sage, TeX

`C_{24}\rtimes_7D_6`
`% in TeX`

`G:=Group("C24:7D6");`
`// GroupNames label`

`G:=SmallGroup(288,771);`
`// by ID`

`G=gap.SmallGroup(288,771);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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