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## G = D24.S3order 288 = 25·32

### 2nd non-split extension by D24 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — D24.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×D24 — D24.S3
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — D24.S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D24.S3
G = < a,b,c,d | a24=b2=c3=1, d2=a12, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 238 in 57 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24, C24, Dic6, D12, C3×D4, C3×Q8, SD32, C3×Dic3, C3×C12, S3×C6, C3⋊C16, D24, Dic12, C3×D8, C3×Q16, C3×C24, C3×Dic6, C3×D12, D8.S3, C8.6D6, C24.S3, C3×D24, C3×Dic12, D24.S3
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊D4, SD32, S32, D4⋊S3, D6⋊S3, D8.S3, C8.6D6, C322D8, D24.S3

Smallest permutation representation of D24.S3
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 45 13 33)(2 28 14 40)(3 35 15 47)(4 42 16 30)(5 25 17 37)(6 32 18 44)(7 39 19 27)(8 46 20 34)(9 29 21 41)(10 36 22 48)(11 43 23 31)(12 26 24 38)(49 88 61 76)(50 95 62 83)(51 78 63 90)(52 85 64 73)(53 92 65 80)(54 75 66 87)(55 82 67 94)(56 89 68 77)(57 96 69 84)(58 79 70 91)(59 86 71 74)(60 93 72 81)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A ··· 24H order 1 2 2 3 3 3 4 4 6 6 6 6 6 8 8 12 12 12 12 12 12 16 16 16 16 24 ··· 24 size 1 1 24 2 2 4 2 24 2 2 4 24 24 2 2 4 4 4 4 24 24 18 18 18 18 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + - - + image C1 C2 C2 C2 S3 S3 D4 D6 D8 C3⋊D4 SD32 S32 D4⋊S3 D6⋊S3 D8.S3 C8.6D6 C32⋊2D8 D24.S3 kernel D24.S3 C24.S3 C3×D24 C3×Dic12 D24 Dic12 C3×C12 C24 C3×C6 C12 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 4 4 1 2 1 2 2 2 4

Matrix representation of D24.S3 in GL6(𝔽97)

 90 7 0 0 0 0 90 90 0 0 0 0 0 0 96 96 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 48 92 0 0 0 0 92 49 0 0 0 0 0 0 96 96 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0
,
 21 25 0 0 0 0 25 76 0 0 0 0 0 0 96 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(97))| [90,90,0,0,0,0,7,90,0,0,0,0,0,0,96,1,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[48,92,0,0,0,0,92,49,0,0,0,0,0,0,96,0,0,0,0,0,96,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[21,25,0,0,0,0,25,76,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D24.S3 in GAP, Magma, Sage, TeX

`D_{24}.S_3`
`% in TeX`

`G:=Group("D24.S3");`
`// GroupNames label`

`G:=SmallGroup(288,195);`
`// by ID`

`G=gap.SmallGroup(288,195);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=b^2=c^3=1,d^2=a^12,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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