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## G = D4⋊2D15order 240 = 24·3·5

### The semidirect product of D4 and D15 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D4⋊2D15
 Chief series C1 — C5 — C15 — C30 — D30 — C4×D15 — D4⋊2D15
 Lower central C15 — C30 — D4⋊2D15
 Upper central C1 — C2 — D4

Generators and relations for D42D15
G = < a,b,c,d | a4=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 352 in 80 conjugacy classes, 35 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×2], C22, C5, S3, C6, C6 [×2], C2×C4 [×3], D4, D4 [×2], Q8, D5, C10, C10 [×2], Dic3 [×3], C12, D6, C2×C6 [×2], C15, C4○D4, Dic5 [×3], C20, D10, C2×C10 [×2], Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, D15, C30, C30 [×2], Dic10, C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4, D42S3, Dic15, Dic15 [×2], C60, D30, C2×C30 [×2], D42D5, Dic30, C4×D15, C2×Dic15 [×2], C157D4 [×2], D4×C15, D42D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, D15, C22×D5, D42S3, D30 [×3], D42D5, C22×D15, D42D15

Smallest permutation representation of D42D15
On 120 points
Generators in S120
```(1 88 17 62)(2 89 18 63)(3 90 19 64)(4 76 20 65)(5 77 21 66)(6 78 22 67)(7 79 23 68)(8 80 24 69)(9 81 25 70)(10 82 26 71)(11 83 27 72)(12 84 28 73)(13 85 29 74)(14 86 30 75)(15 87 16 61)(31 110 58 103)(32 111 59 104)(33 112 60 105)(34 113 46 91)(35 114 47 92)(36 115 48 93)(37 116 49 94)(38 117 50 95)(39 118 51 96)(40 119 52 97)(41 120 53 98)(42 106 54 99)(43 107 55 100)(44 108 56 101)(45 109 57 102)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 91)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 73)(32 74)(33 75)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 87)(47 88)(48 89)(49 90)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)
(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 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 50)(32 49)(33 48)(34 47)(35 46)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 84)(77 83)(78 82)(79 81)(85 90)(86 89)(87 88)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)```

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D15 D30 D30 D4⋊2S3 D4⋊2D5 D4⋊2D15 kernel D4⋊2D15 Dic30 C4×D15 C2×Dic15 C15⋊7D4 D4×C15 C5×D4 C3×D4 C20 C2×C10 C15 C12 C2×C6 D4 C4 C22 C5 C3 C1 # reps 1 1 1 2 2 1 1 2 1 2 2 2 4 4 4 8 1 2 4

Matrix representation of D42D15 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 50
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 50 0 0 0 0 11 0
,
 27 5 0 0 0 0 56 33 0 0 0 0 0 0 44 17 0 0 0 0 44 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 11 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60

`G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0],[27,56,0,0,0,0,5,33,0,0,0,0,0,0,44,44,0,0,0,0,17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,11,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;`

D42D15 in GAP, Magma, Sage, TeX

`D_4\rtimes_2D_{15}`
`% in TeX`

`G:=Group("D4:2D15");`
`// GroupNames label`

`G:=SmallGroup(240,180);`
`// by ID`

`G=gap.SmallGroup(240,180);`
`# by ID`

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

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

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