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## G = C24.3D6order 288 = 25·32

### 3rd non-split extension by C24 of 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 — S3×C12 — S3×Dic6 — C24.3D6
 Lower central C32 — C3×C6 — C3×C12 — C24.3D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C24.3D6
G = < a,b,c | a24=1, b6=c2=a12, bab-1=a11, cac-1=a-1, cbc-1=b5 >

Subgroups: 530 in 129 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3, Dic3 [×7], C12 [×2], C12 [×3], D6, D6, C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×8], C4×S3, C4×S3 [×2], D12, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3 [×2], C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, C24⋊C2, Dic12 [×4], D4.S3, C3⋊Q16, C3×M4(2), C3×SD16, C2×Dic6, C4○D12, D42S3, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3 [×2], D6⋊S3, C322Q8, C3×Dic6, S3×C12, C3×D12, C324Q8 [×2], C8.D6, D4.D6, D12.S3, C323Q16, C3×C8⋊S3, C3×C24⋊C2, C325Q16, S3×Dic6, D125S3, C24.3D6
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, D4.D6, S3×D12, C24.3D6

Smallest permutation representation of C24.3D6
On 96 points
Generators in S96
```(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 44 5 40 9 36 13 32 17 28 21 48)(2 31 6 27 10 47 14 43 18 39 22 35)(3 42 7 38 11 34 15 30 19 26 23 46)(4 29 8 25 12 45 16 41 20 37 24 33)(49 83 69 87 65 91 61 95 57 75 53 79)(50 94 70 74 66 78 62 82 58 86 54 90)(51 81 71 85 67 89 63 93 59 73 55 77)(52 92 72 96 68 76 64 80 60 84 56 88)
(1 86 13 74)(2 85 14 73)(3 84 15 96)(4 83 16 95)(5 82 17 94)(6 81 18 93)(7 80 19 92)(8 79 20 91)(9 78 21 90)(10 77 22 89)(11 76 23 88)(12 75 24 87)(25 65 37 53)(26 64 38 52)(27 63 39 51)(28 62 40 50)(29 61 41 49)(30 60 42 72)(31 59 43 71)(32 58 44 70)(33 57 45 69)(34 56 46 68)(35 55 47 67)(36 54 48 66)```

`G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,5,40,9,36,13,32,17,28,21,48)(2,31,6,27,10,47,14,43,18,39,22,35)(3,42,7,38,11,34,15,30,19,26,23,46)(4,29,8,25,12,45,16,41,20,37,24,33)(49,83,69,87,65,91,61,95,57,75,53,79)(50,94,70,74,66,78,62,82,58,86,54,90)(51,81,71,85,67,89,63,93,59,73,55,77)(52,92,72,96,68,76,64,80,60,84,56,88), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,65,37,53)(26,64,38,52)(27,63,39,51)(28,62,40,50)(29,61,41,49)(30,60,42,72)(31,59,43,71)(32,58,44,70)(33,57,45,69)(34,56,46,68)(35,55,47,67)(36,54,48,66)>;`

`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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,5,40,9,36,13,32,17,28,21,48)(2,31,6,27,10,47,14,43,18,39,22,35)(3,42,7,38,11,34,15,30,19,26,23,46)(4,29,8,25,12,45,16,41,20,37,24,33)(49,83,69,87,65,91,61,95,57,75,53,79)(50,94,70,74,66,78,62,82,58,86,54,90)(51,81,71,85,67,89,63,93,59,73,55,77)(52,92,72,96,68,76,64,80,60,84,56,88), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,65,37,53)(26,64,38,52)(27,63,39,51)(28,62,40,50)(29,61,41,49)(30,60,42,72)(31,59,43,71)(32,58,44,70)(33,57,45,69)(34,56,46,68)(35,55,47,67)(36,54,48,66) );`

`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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,44,5,40,9,36,13,32,17,28,21,48),(2,31,6,27,10,47,14,43,18,39,22,35),(3,42,7,38,11,34,15,30,19,26,23,46),(4,29,8,25,12,45,16,41,20,37,24,33),(49,83,69,87,65,91,61,95,57,75,53,79),(50,94,70,74,66,78,62,82,58,86,54,90),(51,81,71,85,67,89,63,93,59,73,55,77),(52,92,72,96,68,76,64,80,60,84,56,88)], [(1,86,13,74),(2,85,14,73),(3,84,15,96),(4,83,16,95),(5,82,17,94),(6,81,18,93),(7,80,19,92),(8,79,20,91),(9,78,21,90),(10,77,22,89),(11,76,23,88),(12,75,24,87),(25,65,37,53),(26,64,38,52),(27,63,39,51),(28,62,40,50),(29,61,41,49),(30,60,42,72),(31,59,43,71),(32,58,44,70),(33,57,45,69),(34,56,46,68),(35,55,47,67),(36,54,48,66)])`

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A ··· 24H 24I 24J order 1 2 2 2 3 3 3 4 4 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 2 2 4 2 6 12 36 36 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 D4.D6 S3×D12 C24.3D6 kernel C24.3D6 D12.S3 C32⋊3Q16 C3×C8⋊S3 C3×C24⋊C2 C32⋊5Q16 S3×Dic6 D12⋊5S3 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 C24.3D6 in GL4(𝔽73) generated by

 23 55 0 0 18 5 0 0 17 38 50 55 32 31 18 68
,
 13 28 2 1 13 28 1 2 66 6 55 50 66 7 55 50
,
 12 10 0 0 22 61 0 0 26 50 10 61 42 62 51 63
`G:=sub<GL(4,GF(73))| [23,18,17,32,55,5,38,31,0,0,50,18,0,0,55,68],[13,13,66,66,28,28,6,7,2,1,55,55,1,2,50,50],[12,22,26,42,10,61,50,62,0,0,10,51,0,0,61,63] >;`

C24.3D6 in GAP, Magma, Sage, TeX

`C_{24}._3D_6`
`% in TeX`

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

`G:=SmallGroup(288,448);`
`// by ID`

`G=gap.SmallGroup(288,448);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,58,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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