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G = D15⋊D8order 480 = 25·3·5

The semidirect product of D15 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D15⋊D8
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C20⋊D6 — D15⋊D8
 Lower central C15 — C30 — C60 — D15⋊D8
 Upper central C1 — C2 — C4 — D4

Generators and relations for D15⋊D8
G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1228 in 152 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, D15, C30, C30, C2×D8, C52C8, C40, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×D4, C22×D5, S3×C8, D24, D4⋊S3, D4⋊S3, C3×D8, S3×D4, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C8×D5, D40, D4⋊D5, D4⋊D5, C5×D8, D4×D5, S3×D8, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, C4×D15, D60, C157D4, D4×C15, C2×S3×D5, C22×D15, D5×D8, D152C8, C3⋊D40, C5⋊D24, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D4×D15, D15⋊D8
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, D10⋊D6, D15⋊D8

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

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

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

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

45 conjugacy classes

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

45 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 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D8 D10 D10 D10 S3×D4 S3×D5 D4×D5 S3×D8 C2×S3×D5 D5×D8 D10⋊D6 D15⋊D8 kernel D15⋊D8 D15⋊2C8 C3⋊D40 C5⋊D24 C3×D4⋊D5 C5×D4⋊S3 C20⋊D6 D4×D15 D4⋊D5 Dic15 D30 D4⋊S3 C5⋊2C8 D20 C5×D4 D15 C3⋊C8 D12 C3×D4 C10 D4 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of D15⋊D8 in GL6(𝔽241)

 240 189 0 0 0 0 52 52 0 0 0 0 0 0 1 54 0 0 0 0 174 239 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 52 0 0 0 0 0 240 0 0 0 0 0 0 240 187 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 174 240 0 0 0 0 0 0 0 22 0 0 0 0 230 22
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 174 240 0 0 0 0 0 0 1 0 0 0 0 0 1 240

`G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,1,174,0,0,0,0,54,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,52,240,0,0,0,0,0,0,240,0,0,0,0,0,187,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,0,230,0,0,0,0,22,22],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,240] >;`

D15⋊D8 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,557);`
`// by ID`

`G=gap.SmallGroup(480,557);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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