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G = D24⋊D5order 480 = 25·3·5

2nd semidirect product of D24 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D24⋊D5
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — D24⋊D5
 Lower central C15 — C30 — C60 — D24⋊D5
 Upper central C1 — C2 — C4 — C8

Generators and relations for D24⋊D5
G = < a,b,c,d | a24=b2=c5=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a12b, dcd=c-1 >

Subgroups: 988 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C24, C24, Dic6, C4×S3, D12 [×2], D12 [×2], C3⋊D4, C2×C12, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C24⋊C2 [×2], D24, D24, C3×M4(2), C2×D12, C4○D12, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10 [×2], D30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, C8⋊D6, C3×C52C8, C120, S3×Dic5, C15⋊D4, C5⋊D12, D5×C12, C5×D12 [×2], Dic30, D60, C2×S3×D5, D8⋊D5, C5⋊D24, D12.D5, C3×C8⋊D5, C5×D24, C24⋊D5, D125D5, D5×D12, D24⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C22×D5, C2×D12, S3×D5, D4×D5, C8⋊D6, C2×S3×D5, D8⋊D5, D5×D12, D24⋊D5

Smallest permutation representation of D24⋊D5
On 120 points
Generators in S120
```(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 24)(20 23)(21 22)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(45 48)(46 47)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(89 96)(90 95)(91 94)(92 93)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(113 120)(114 119)(115 118)(116 117)
(1 108 96 38 70)(2 109 73 39 71)(3 110 74 40 72)(4 111 75 41 49)(5 112 76 42 50)(6 113 77 43 51)(7 114 78 44 52)(8 115 79 45 53)(9 116 80 46 54)(10 117 81 47 55)(11 118 82 48 56)(12 119 83 25 57)(13 120 84 26 58)(14 97 85 27 59)(15 98 86 28 60)(16 99 87 29 61)(17 100 88 30 62)(18 101 89 31 63)(19 102 90 32 64)(20 103 91 33 65)(21 104 92 34 66)(22 105 93 35 67)(23 106 94 36 68)(24 107 95 37 69)
(1 70)(2 59)(3 72)(4 61)(5 50)(6 63)(7 52)(8 65)(9 54)(10 67)(11 56)(12 69)(13 58)(14 71)(15 60)(16 49)(17 62)(18 51)(19 64)(20 53)(21 66)(22 55)(23 68)(24 57)(25 107)(26 120)(27 109)(28 98)(29 111)(30 100)(31 113)(32 102)(33 115)(34 104)(35 117)(36 106)(37 119)(38 108)(39 97)(40 110)(41 99)(42 112)(43 101)(44 114)(45 103)(46 116)(47 105)(48 118)(73 85)(75 87)(77 89)(79 91)(81 93)(83 95)```

`G:=sub<Sym(120)| (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,48)(46,47)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(89,96)(90,95)(91,94)(92,93)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(113,120)(114,119)(115,118)(116,117), (1,108,96,38,70)(2,109,73,39,71)(3,110,74,40,72)(4,111,75,41,49)(5,112,76,42,50)(6,113,77,43,51)(7,114,78,44,52)(8,115,79,45,53)(9,116,80,46,54)(10,117,81,47,55)(11,118,82,48,56)(12,119,83,25,57)(13,120,84,26,58)(14,97,85,27,59)(15,98,86,28,60)(16,99,87,29,61)(17,100,88,30,62)(18,101,89,31,63)(19,102,90,32,64)(20,103,91,33,65)(21,104,92,34,66)(22,105,93,35,67)(23,106,94,36,68)(24,107,95,37,69), (1,70)(2,59)(3,72)(4,61)(5,50)(6,63)(7,52)(8,65)(9,54)(10,67)(11,56)(12,69)(13,58)(14,71)(15,60)(16,49)(17,62)(18,51)(19,64)(20,53)(21,66)(22,55)(23,68)(24,57)(25,107)(26,120)(27,109)(28,98)(29,111)(30,100)(31,113)(32,102)(33,115)(34,104)(35,117)(36,106)(37,119)(38,108)(39,97)(40,110)(41,99)(42,112)(43,101)(44,114)(45,103)(46,116)(47,105)(48,118)(73,85)(75,87)(77,89)(79,91)(81,93)(83,95)>;`

`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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(45,48)(46,47)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(89,96)(90,95)(91,94)(92,93)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)(113,120)(114,119)(115,118)(116,117), (1,108,96,38,70)(2,109,73,39,71)(3,110,74,40,72)(4,111,75,41,49)(5,112,76,42,50)(6,113,77,43,51)(7,114,78,44,52)(8,115,79,45,53)(9,116,80,46,54)(10,117,81,47,55)(11,118,82,48,56)(12,119,83,25,57)(13,120,84,26,58)(14,97,85,27,59)(15,98,86,28,60)(16,99,87,29,61)(17,100,88,30,62)(18,101,89,31,63)(19,102,90,32,64)(20,103,91,33,65)(21,104,92,34,66)(22,105,93,35,67)(23,106,94,36,68)(24,107,95,37,69), (1,70)(2,59)(3,72)(4,61)(5,50)(6,63)(7,52)(8,65)(9,54)(10,67)(11,56)(12,69)(13,58)(14,71)(15,60)(16,49)(17,62)(18,51)(19,64)(20,53)(21,66)(22,55)(23,68)(24,57)(25,107)(26,120)(27,109)(28,98)(29,111)(30,100)(31,113)(32,102)(33,115)(34,104)(35,117)(36,106)(37,119)(38,108)(39,97)(40,110)(41,99)(42,112)(43,101)(44,114)(45,103)(46,116)(47,105)(48,118)(73,85)(75,87)(77,89)(79,91)(81,93)(83,95) );`

`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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,24),(20,23),(21,22),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(45,48),(46,47),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(89,96),(90,95),(91,94),(92,93),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105),(113,120),(114,119),(115,118),(116,117)], [(1,108,96,38,70),(2,109,73,39,71),(3,110,74,40,72),(4,111,75,41,49),(5,112,76,42,50),(6,113,77,43,51),(7,114,78,44,52),(8,115,79,45,53),(9,116,80,46,54),(10,117,81,47,55),(11,118,82,48,56),(12,119,83,25,57),(13,120,84,26,58),(14,97,85,27,59),(15,98,86,28,60),(16,99,87,29,61),(17,100,88,30,62),(18,101,89,31,63),(19,102,90,32,64),(20,103,91,33,65),(21,104,92,34,66),(22,105,93,35,67),(23,106,94,36,68),(24,107,95,37,69)], [(1,70),(2,59),(3,72),(4,61),(5,50),(6,63),(7,52),(8,65),(9,54),(10,67),(11,56),(12,69),(13,58),(14,71),(15,60),(16,49),(17,62),(18,51),(19,64),(20,53),(21,66),(22,55),(23,68),(24,57),(25,107),(26,120),(27,109),(28,98),(29,111),(30,100),(31,113),(32,102),(33,115),(34,104),(35,117),(36,106),(37,119),(38,108),(39,97),(40,110),(41,99),(42,112),(43,101),(44,114),(45,103),(46,116),(47,105),(48,118),(73,85),(75,87),(77,89),(79,91),(81,93),(83,95)])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 12C 15A 15B 20A 20B 24A 24B 24C 24D 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 8 8 10 10 10 10 10 10 12 12 12 15 15 20 20 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 10 12 12 60 2 2 10 60 2 2 2 20 4 20 2 2 24 24 24 24 2 2 20 4 4 4 4 4 4 20 20 4 4 4 4 4 4 4 4 4 4 4 ··· 4

51 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 D4 D4 D5 D6 D6 D6 D10 D10 D12 D12 C8⋊C22 S3×D5 D4×D5 C8⋊D6 C2×S3×D5 D8⋊D5 D5×D12 D24⋊D5 kernel D24⋊D5 C5⋊D24 D12.D5 C3×C8⋊D5 C5×D24 C24⋊D5 D12⋊5D5 D5×D12 C8⋊D5 C3×Dic5 C6×D5 D24 C5⋊2C8 C40 C4×D5 C24 D12 Dic5 D10 C15 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 4 2 2 1 2 2 2 2 4 4 8

Matrix representation of D24⋊D5 in GL4(𝔽241) generated by

 2 204 186 174 37 239 67 55 55 67 57 30 174 186 211 184
,
 239 37 184 211 204 2 30 57 186 174 2 204 67 55 37 239
,
 0 1 0 0 240 189 0 0 0 0 0 1 0 0 240 189
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(241))| [2,37,55,174,204,239,67,186,186,67,57,211,174,55,30,184],[239,204,186,67,37,2,174,55,184,30,2,37,211,57,204,239],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

D24⋊D5 in GAP, Magma, Sage, TeX

`D_{24}\rtimes D_5`
`% in TeX`

`G:=Group("D24:D5");`
`// GroupNames label`

`G:=SmallGroup(480,326);`
`// by ID`

`G=gap.SmallGroup(480,326);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,142,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^24=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;`
`// generators/relations`

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