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## G = C24⋊D10order 480 = 25·3·5

### 1st semidirect product of C24 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C24⋊D10
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — C24⋊D10
 Lower central C15 — C30 — C60 — C24⋊D10
 Upper central C1 — C2 — C4 — C8

Generators and relations for C24⋊D10
G = < a,b,c | a24=b10=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >

Subgroups: 1052 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 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C24, C24, Dic6, C4×S3, D12, D12 [×3], C3⋊D4, C2×C12, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20 [×3], C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, D24 [×2], C3×M4(2), C2×D12, C4○D12, C5×Dic3, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30 [×2], C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C8⋊D6, C3×C52C8, C120, D30.C2, C3⋊D20, C5⋊D12, D5×C12, C5×Dic6, C5×D12, D60 [×2], C2×S3×D5, D40⋊C2, C5⋊D24, Dic6⋊D5, C3×C8⋊D5, C5×C24⋊C2, D120, C12.28D10, D5×D12, C24⋊D10
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, D40⋊C2, D5×D12, C24⋊D10

Smallest permutation representation of C24⋊D10
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 39 84 105 51 7 33 90 99 57)(2 26 85 116 52 18 34 77 100 68)(3 37 86 103 53 5 35 88 101 55)(4 48 87 114 54 16 36 75 102 66)(6 46 89 112 56 14 38 73 104 64)(8 44 91 110 58 12 40 95 106 62)(9 31 92 97 59 23 41 82 107 49)(10 42 93 108 60)(11 29 94 119 61 21 43 80 109 71)(13 27 96 117 63 19 45 78 111 69)(15 25 74 115 65 17 47 76 113 67)(20 32 79 98 70 24 28 83 118 50)(22 30 81 120 72)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(73 77)(74 76)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 8A 8B 10A 10B 10C 10D 12A 12B 12C 15A 15B 20A 20B 20C 20D 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 12 12 12 15 15 20 20 20 20 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 10 12 60 60 2 2 10 12 2 2 2 20 4 20 2 2 24 24 2 2 20 4 4 4 4 24 24 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 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 D10 D12 D12 C8⋊C22 S3×D5 D4×D5 C8⋊D6 C2×S3×D5 D40⋊C2 D5×D12 C24⋊D10 kernel C24⋊D10 C5⋊D24 Dic6⋊D5 C3×C8⋊D5 C5×C24⋊C2 D120 C12.28D10 D5×D12 C8⋊D5 C3×Dic5 C6×D5 C24⋊C2 C5⋊2C8 C40 C4×D5 C24 Dic6 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 2 2 2 2 1 2 2 2 2 4 4 8

Matrix representation of C24⋊D10 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 143 100 2 2 0 0 141 43 239 0 0 0 170 71 98 141 0 0 170 99 100 198
,
 0 189 0 0 0 0 51 190 0 0 0 0 0 0 99 198 0 0 0 0 99 142 0 0 0 0 141 43 240 0 0 0 143 100 1 1
,
 51 189 0 0 0 0 50 190 0 0 0 0 0 0 99 198 0 0 0 0 99 142 0 0 0 0 0 0 1 0 0 0 0 0 240 240

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,143,141,170,170,0,0,100,43,71,99,0,0,2,239,98,100,0,0,2,0,141,198],[0,51,0,0,0,0,189,190,0,0,0,0,0,0,99,99,141,143,0,0,198,142,43,100,0,0,0,0,240,1,0,0,0,0,0,1],[51,50,0,0,0,0,189,190,0,0,0,0,0,0,99,99,0,0,0,0,198,142,0,0,0,0,0,0,1,240,0,0,0,0,0,240] >;`

C24⋊D10 in GAP, Magma, Sage, TeX

`C_{24}\rtimes D_{10}`
`% in TeX`

`G:=Group("C24:D10");`
`// GroupNames label`

`G:=SmallGroup(480,325);`
`// by ID`

`G=gap.SmallGroup(480,325);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,58,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^10=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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