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## G = D15⋊D7order 420 = 22·3·5·7

### The semidirect product of D15 and D7 acting via D7/C7=C2

Aliases: D15⋊D7, D35⋊S3, D21⋊D5, C353D6, C153D14, C213D10, C1054C22, C53(S3×D7), C33(D5×D7), C73(S3×D5), (C7×D15)⋊2C2, (C5×D21)⋊2C2, (C3×D35)⋊2C2, SmallGroup(420,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — D15⋊D7
 Chief series C1 — C7 — C35 — C105 — C5×D21 — D15⋊D7
 Lower central C105 — D15⋊D7
 Upper central C1

Generators and relations for D15⋊D7
G = < a,b,c,d | a21=b2=c5=d2=1, bab=a-1, ac=ca, dad=a8, bc=cb, dbd=a7b, dcd=c-1 >

15C2
21C2
35C2
105C22
5S3
7S3
35C6
3D5
7D5
21C10
3D7
5D7
15C14
35D6
21D10
15D14

Smallest permutation representation of D15⋊D7
On 105 points
Generators in S105
```(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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 42)(43 50)(44 49)(45 48)(46 47)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(64 66)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)
(1 47 42 96 76)(2 48 22 97 77)(3 49 23 98 78)(4 50 24 99 79)(5 51 25 100 80)(6 52 26 101 81)(7 53 27 102 82)(8 54 28 103 83)(9 55 29 104 84)(10 56 30 105 64)(11 57 31 85 65)(12 58 32 86 66)(13 59 33 87 67)(14 60 34 88 68)(15 61 35 89 69)(16 62 36 90 70)(17 63 37 91 71)(18 43 38 92 72)(19 44 39 93 73)(20 45 40 94 74)(21 46 41 95 75)
(1 76)(2 84)(3 71)(4 79)(5 66)(6 74)(7 82)(8 69)(9 77)(10 64)(11 72)(12 80)(13 67)(14 75)(15 83)(16 70)(17 78)(18 65)(19 73)(20 81)(21 68)(22 29)(23 37)(25 32)(26 40)(28 35)(31 38)(34 41)(43 85)(44 93)(45 101)(46 88)(47 96)(48 104)(49 91)(50 99)(51 86)(52 94)(53 102)(54 89)(55 97)(56 105)(57 92)(58 100)(59 87)(60 95)(61 103)(62 90)(63 98)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 6 7A 7B 7C 10A 10B 14A 14B 14C 15A 15B 21A 21B 21C 35A ··· 35F 105A ··· 105L order 1 2 2 2 3 5 5 6 7 7 7 10 10 14 14 14 15 15 21 21 21 35 ··· 35 105 ··· 105 size 1 15 21 35 2 2 2 70 2 2 2 42 42 30 30 30 4 4 4 4 4 4 ··· 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D6 D7 D10 D14 S3×D5 S3×D7 D5×D7 D15⋊D7 kernel D15⋊D7 C3×D35 C5×D21 C7×D15 D35 D21 C35 D15 C21 C15 C7 C5 C3 C1 # reps 1 1 1 1 1 2 1 3 2 3 2 3 6 12

Matrix representation of D15⋊D7 in GL6(𝔽211)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 81 100 0 0 0 0 192 0 0 0 0 0 0 0 209 56 0 0 0 0 64 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 100 0 0 0 0 19 0 0 0 0 0 0 0 210 0 0 0 0 0 64 1
,
 0 1 0 0 0 0 210 32 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 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 210 56 0 0 0 0 0 1

`G:=sub<GL(6,GF(211))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,81,192,0,0,0,0,100,0,0,0,0,0,0,0,209,64,0,0,0,0,56,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,19,0,0,0,0,100,0,0,0,0,0,0,0,210,64,0,0,0,0,0,1],[0,210,0,0,0,0,1,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,210,0,0,0,0,0,56,1] >;`

D15⋊D7 in GAP, Magma, Sage, TeX

`D_{15}\rtimes D_7`
`% in TeX`

`G:=Group("D15:D7");`
`// GroupNames label`

`G:=SmallGroup(420,30);`
`// by ID`

`G=gap.SmallGroup(420,30);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,-7,122,67,488,9004]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^21=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^8,b*c=c*b,d*b*d=a^7*b,d*c*d=c^-1>;`
`// generators/relations`

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