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## G = Q8⋊3D15order 240 = 24·3·5

### The semidirect product of Q8 and D15 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Q8⋊3D15
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15 — Q8⋊3D15
 Lower central C15 — C30 — Q8⋊3D15
 Upper central C1 — C2 — Q8

Generators and relations for Q83D15
G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 432 in 80 conjugacy classes, 35 normal (14 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3, C12 [×3], D6 [×3], C15, C4○D4, Dic5, C20 [×3], D10 [×3], C4×S3 [×3], D12 [×3], C3×Q8, D15 [×3], C30, C4×D5 [×3], D20 [×3], C5×Q8, Q83S3, Dic15, C60 [×3], D30 [×3], Q82D5, C4×D15 [×3], D60 [×3], Q8×C15, Q83D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, D15, C22×D5, Q83S3, D30 [×3], Q82D5, C22×D15, Q83D15

Smallest permutation representation of Q83D15
On 120 points
Generators in S120
```(1 58 16 32)(2 59 17 33)(3 60 18 34)(4 46 19 35)(5 47 20 36)(6 48 21 37)(7 49 22 38)(8 50 23 39)(9 51 24 40)(10 52 25 41)(11 53 26 42)(12 54 27 43)(13 55 28 44)(14 56 29 45)(15 57 30 31)(61 98 87 120)(62 99 88 106)(63 100 89 107)(64 101 90 108)(65 102 76 109)(66 103 77 110)(67 104 78 111)(68 105 79 112)(69 91 80 113)(70 92 81 114)(71 93 82 115)(72 94 83 116)(73 95 84 117)(74 96 85 118)(75 97 86 119)
(1 88 16 62)(2 89 17 63)(3 90 18 64)(4 76 19 65)(5 77 20 66)(6 78 21 67)(7 79 22 68)(8 80 23 69)(9 81 24 70)(10 82 25 71)(11 83 26 72)(12 84 27 73)(13 85 28 74)(14 86 29 75)(15 87 30 61)(31 120 57 98)(32 106 58 99)(33 107 59 100)(34 108 60 101)(35 109 46 102)(36 110 47 103)(37 111 48 104)(38 112 49 105)(39 113 50 91)(40 114 51 92)(41 115 52 93)(42 116 53 94)(43 117 54 95)(44 118 55 96)(45 119 56 97)
(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 31)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 60)(29 59)(30 58)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 120)(89 119)(90 118)```

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

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

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

Q83D15 is a maximal subgroup of
D60⋊C22  C60.C23  D20.16D6  D12.D10  Q83D30  D4.5D30  Q16⋊D15  D1208C2  C30.33C24  D5×Q83S3  S3×Q82D5  D2017D6  Q8.15D30  C4○D4×D15  D48D30
Q83D15 is a maximal quotient of
C4.Dic30  C4⋊C47D15  D6011C4  D30.29D4  C4⋊D60  C4⋊C4⋊D15  Q8×Dic15  D307Q8  C60.23D4

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 C4○D4 D10 D15 D30 Q8⋊3S3 Q8⋊2D5 Q8⋊3D15 kernel Q8⋊3D15 C4×D15 D60 Q8×C15 C5×Q8 C3×Q8 C20 C15 C12 Q8 C4 C5 C3 C1 # reps 1 3 3 1 1 2 3 2 6 4 12 1 2 4

Matrix representation of Q83D15 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 0
,
 1 0 0 0 0 1 0 0 0 0 0 50 0 0 50 0
,
 9 56 0 0 5 38 0 0 0 0 1 0 0 0 0 1
,
 52 5 0 0 45 9 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,50,0,0,50,0],[9,5,0,0,56,38,0,0,0,0,1,0,0,0,0,1],[52,45,0,0,5,9,0,0,0,0,0,1,0,0,1,0] >;`

Q83D15 in GAP, Magma, Sage, TeX

`Q_8\rtimes_3D_{15}`
`% in TeX`

`G:=Group("Q8:3D15");`
`// GroupNames label`

`G:=SmallGroup(240,182);`
`// by ID`

`G=gap.SmallGroup(240,182);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,964,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^15=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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