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G = D8⋊D15order 480 = 25·3·5

2nd semidirect product of D8 and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D8⋊D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — D4×D15 — D8⋊D15
 Lower central C15 — C30 — C60 — D8⋊D15
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8⋊D15
G = < a,b,c,d | a4=b2=c30=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >

Subgroups: 1012 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, D15 [×2], C30, C30 [×2], C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, Dic15, Dic15, C60, D30, D30 [×3], C2×C30 [×2], C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D8⋊S3, C153C8, C120, Dic30, C4×D15, D60, C2×Dic15, C157D4 [×2], D4×C15 [×2], C22×D15, D8⋊D5, C40⋊S3, C24⋊D5, D4⋊D15, D4.D15, C15×D8, D4×D15, D42D15, D8⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30 [×3], D4×D5, D8⋊S3, C22×D15, D8⋊D5, D4×D15, D8⋊D15

Smallest permutation representation of D8⋊D15
On 120 points
Generators in S120
```(1 73 19 88)(2 89 20 74)(3 75 21 90)(4 61 22 76)(5 77 23 62)(6 63 24 78)(7 79 25 64)(8 65 26 80)(9 81 27 66)(10 67 28 82)(11 83 29 68)(12 69 30 84)(13 85 16 70)(14 71 17 86)(15 87 18 72)(31 46 115 100)(32 101 116 47)(33 48 117 102)(34 103 118 49)(35 50 119 104)(36 105 120 51)(37 52 91 106)(38 107 92 53)(39 54 93 108)(40 109 94 55)(41 56 95 110)(42 111 96 57)(43 58 97 112)(44 113 98 59)(45 60 99 114)
(1 92)(2 108)(3 94)(4 110)(5 96)(6 112)(7 98)(8 114)(9 100)(10 116)(11 102)(12 118)(13 104)(14 120)(15 106)(16 50)(17 36)(18 52)(19 38)(20 54)(21 40)(22 56)(23 42)(24 58)(25 44)(26 60)(27 46)(28 32)(29 48)(30 34)(31 66)(33 68)(35 70)(37 72)(39 74)(41 76)(43 78)(45 80)(47 82)(49 84)(51 86)(53 88)(55 90)(57 62)(59 64)(61 95)(63 97)(65 99)(67 101)(69 103)(71 105)(73 107)(75 109)(77 111)(79 113)(81 115)(83 117)(85 119)(87 91)(89 93)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(15 30)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(119 120)```

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

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

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

60 conjugacy classes

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

60 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 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D15 D30 D30 C8⋊C22 S3×D4 D4×D5 D8⋊S3 D8⋊D5 D4×D15 D8⋊D15 kernel D8⋊D15 C40⋊S3 C24⋊D5 D4⋊D15 D4.D15 C15×D8 D4×D15 D4⋊2D15 C5×D8 Dic15 D30 C3×D8 C40 C5×D4 C24 C3×D4 D8 C8 D4 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 4 4 4 8 1 1 2 2 4 4 8

Matrix representation of D8⋊D15 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 239 0 0 0 0 1 0 239 0 0 1 0 240 0 0 0 0 1 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 94 53 147 188 0 0 188 147 53 94 0 0 47 147 147 188 0 0 94 194 53 94
,
 52 240 0 0 0 0 53 240 0 0 0 0 0 0 240 1 2 239 0 0 240 0 2 0 0 0 0 0 1 240 0 0 0 0 1 0
,
 0 51 0 0 0 0 52 0 0 0 0 0 0 0 240 0 2 0 0 0 240 1 2 239 0 0 0 0 1 0 0 0 0 0 1 240

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,239,0,240,0,0,0,0,239,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,94,188,47,94,0,0,53,147,147,194,0,0,147,53,147,53,0,0,188,94,188,94],[52,53,0,0,0,0,240,240,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,2,2,1,1,0,0,239,0,240,0],[0,52,0,0,0,0,51,0,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,2,2,1,1,0,0,0,239,0,240] >;`

D8⋊D15 in GAP, Magma, Sage, TeX

`D_8\rtimes D_{15}`
`% in TeX`

`G:=Group("D8:D15");`
`// GroupNames label`

`G:=SmallGroup(480,876);`
`// by ID`

`G=gap.SmallGroup(480,876);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,346,185,80,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^30=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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