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## G = D20⋊16D6order 480 = 25·3·5

### 10th semidirect product of D20 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊16D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — D20⋊16D6
 Lower central C15 — C30 — D20⋊16D6
 Upper central C1 — C2 — Q8

Generators and relations for D2016D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a9, ad=da, cbc-1=a18b, dbd=a10b, dcd=c-1 >

Subgroups: 1660 in 328 conjugacy classes, 110 normal (24 characteristic)
C1, C2, C2 [×8], C3, C4 [×3], C4 [×5], C22 [×13], C5, S3 [×5], C6, C6 [×3], C2×C4 [×16], D4 [×12], Q8, Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×3], Dic3, Dic3 [×3], C12 [×3], C12, D6 [×3], D6 [×7], C2×C6 [×3], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×3], C20 [×3], C20, D10 [×3], D10 [×7], C2×C10 [×3], Dic6 [×3], C4×S3 [×3], C4×S3 [×7], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×3], C3×D5 [×3], D15 [×2], C30, C2×C4○D4, Dic10 [×3], C4×D5 [×3], C4×D5 [×7], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C3×Dic5, Dic15 [×3], C60 [×3], S3×D5 [×6], C6×D5 [×3], S3×C10 [×3], D30, C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, D5×Dic3 [×3], S3×Dic5 [×3], D30.C2, C15⋊D4 [×6], D5×C12 [×3], C3×D20 [×3], S3×C20 [×3], C5×D12 [×3], Dic30 [×3], C4×D15 [×3], Q8×C15, C2×S3×D5 [×3], D5×C4○D4, D205S3 [×3], D125D5 [×3], C4×S3×D5 [×3], C20⋊D6 [×3], C3×Q82D5, C5×Q83S3, Q8×D15, D2016D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D2016D6

Smallest permutation representation of D2016D6
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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 58 98 40 101 62)(2 47 99 29 102 71)(3 56 100 38 103 80)(4 45 81 27 104 69)(5 54 82 36 105 78)(6 43 83 25 106 67)(7 52 84 34 107 76)(8 41 85 23 108 65)(9 50 86 32 109 74)(10 59 87 21 110 63)(11 48 88 30 111 72)(12 57 89 39 112 61)(13 46 90 28 113 70)(14 55 91 37 114 79)(15 44 92 26 115 68)(16 53 93 35 116 77)(17 42 94 24 117 66)(18 51 95 33 118 75)(19 60 96 22 119 64)(20 49 97 31 120 73)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 15A 15B 20A ··· 20F 20G 20H 20I 20J 30A 30B 60A ··· 60F order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 10 10 10 ··· 10 12 12 12 12 12 15 15 20 ··· 20 20 20 20 20 30 30 60 ··· 60 size 1 1 6 6 6 10 10 10 15 15 2 2 2 2 3 3 5 5 30 30 30 2 2 2 20 20 20 2 2 12 ··· 12 4 4 4 10 10 4 4 4 ··· 4 6 6 6 6 4 4 8 ··· 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 S3×D5 S3×C4○D4 C2×S3×D5 D5×C4○D4 D20⋊16D6 kernel D20⋊16D6 D20⋊5S3 D12⋊5D5 C4×S3×D5 C20⋊D6 C3×Q8⋊2D5 C5×Q8⋊3S3 Q8×D15 Q8⋊2D5 Q8⋊3S3 C4×D5 D20 C5×Q8 D15 C4×S3 D12 C3×Q8 Q8 C5 C4 C3 C1 # reps 1 3 3 3 3 1 1 1 1 2 3 3 1 4 6 6 2 2 2 6 4 2

Matrix representation of D2016D6 in GL6(𝔽61)

 43 1 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 11 0 0 0 0 11 0
,
 1 43 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 50 0 0 0 0 11 0
,
 18 43 0 0 0 0 1 43 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 1 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 60 0

`G:=sub<GL(6,GF(61))| [43,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,11,0],[1,0,0,0,0,0,43,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,11,0,0,0,0,50,0],[18,1,0,0,0,0,43,43,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0] >;`

D2016D6 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{16}D_6`
`% in TeX`

`G:=Group("D20:16D6");`
`// GroupNames label`

`G:=SmallGroup(480,1110);`
`// by ID`

`G=gap.SmallGroup(480,1110);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^9,a*d=d*a,c*b*c^-1=a^18*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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