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## G = D6⋊D15order 360 = 23·32·5

### 1st semidirect product of D6 and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — D6⋊D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — D6⋊D15
 Lower central C3×C15 — C3×C30 — D6⋊D15
 Upper central C1 — C2

Generators and relations for D6⋊D15
G = < a,b,c,d | a6=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 444 in 70 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, D6, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, D10, C2×C10, C3⋊D4, C5×S3, C3×D5, D15, C30, C30, C3⋊Dic3, S3×C6, S3×C6, C5⋊D4, C3×C15, Dic15, C6×D5, S3×C10, D30, C2×C30, D6⋊S3, S3×C15, C3×D15, C3×C30, C15⋊D4, C157D4, C3⋊Dic15, S3×C30, C6×D15, D6⋊D15
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3⋊D4, D15, S32, C5⋊D4, S3×D5, D30, D6⋊S3, C15⋊D4, C157D4, S3×D15, D6⋊D15

Smallest permutation representation of D6⋊D15
On 120 points
Generators in S120
```(1 23 6 28 11 18)(2 24 7 29 12 19)(3 25 8 30 13 20)(4 26 9 16 14 21)(5 27 10 17 15 22)(31 58 41 53 36 48)(32 59 42 54 37 49)(33 60 43 55 38 50)(34 46 44 56 39 51)(35 47 45 57 40 52)(61 89 71 84 66 79)(62 90 72 85 67 80)(63 76 73 86 68 81)(64 77 74 87 69 82)(65 78 75 88 70 83)(91 115 96 120 101 110)(92 116 97 106 102 111)(93 117 98 107 103 112)(94 118 99 108 104 113)(95 119 100 109 105 114)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 101)(62 102)(63 103)(64 104)(65 105)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 120)(90 106)
(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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 75)(13 74)(14 73)(15 72)(16 76)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 112)(55 111)(56 110)(57 109)(58 108)(59 107)(60 106)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D 30E ··· 30J 30K ··· 30R order 1 2 2 2 3 3 3 4 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 15 15 15 15 15 ··· 15 30 30 30 30 30 ··· 30 30 ··· 30 size 1 1 6 30 2 2 4 90 2 2 2 2 4 6 6 30 30 2 2 6 6 6 6 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 6 ··· 6

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 S3 S3 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 S32 S3×D5 D6⋊S3 C15⋊D4 S3×D15 D6⋊D15 kernel D6⋊D15 C3⋊Dic15 S3×C30 C6×D15 S3×C10 D30 C3×C15 S3×C6 C30 C3×C6 C15 D6 C32 C6 C3 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 1 2 1 2 4 4

Matrix representation of D6⋊D15 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 41 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 60 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 49 0 0 0 0 12 30
,
 13 26 0 0 0 0 17 48 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 19 12 0 0 0 0 31 42

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,41,1,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,12,0,0,0,0,49,30],[13,17,0,0,0,0,26,48,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,19,31,0,0,0,0,12,42] >;`

D6⋊D15 in GAP, Magma, Sage, TeX

`D_6\rtimes D_{15}`
`% in TeX`

`G:=Group("D6:D15");`
`// GroupNames label`

`G:=SmallGroup(360,80);`
`// by ID`

`G=gap.SmallGroup(360,80);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,201,1444,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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