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## G = D20⋊S3order 240 = 24·3·5

### 3rd semidirect product of D20 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊S3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D20⋊S3
 Lower central C15 — C30 — D20⋊S3
 Upper central C1 — C2 — C4

Generators and relations for D20⋊S3
G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 360 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3 [×2], Dic3, C12, D6, C2×C6 [×2], C15, C4○D4, Dic5, C20, C20 [×2], D10 [×2], D10, Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, C3×D5 [×2], D15, C30, C4×D5 [×3], D20, D20 [×2], C5×Q8, D42S3, C5×Dic3 [×2], Dic15, C60, C6×D5 [×2], D30, Q82D5, D5×Dic3 [×2], C3⋊D20 [×2], C3×D20, C5×Dic6, C4×D15, D20⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, D42S3, S3×D5, Q82D5, C2×S3×D5, D20⋊S3

Smallest permutation representation of D20⋊S3
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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 81)(22 100)(23 99)(24 98)(25 97)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)
(1 23 118)(2 24 119)(3 25 120)(4 26 101)(5 27 102)(6 28 103)(7 29 104)(8 30 105)(9 31 106)(10 32 107)(11 33 108)(12 34 109)(13 35 110)(14 36 111)(15 37 112)(16 38 113)(17 39 114)(18 40 115)(19 21 116)(20 22 117)(41 72 91)(42 73 92)(43 74 93)(44 75 94)(45 76 95)(46 77 96)(47 78 97)(48 79 98)(49 80 99)(50 61 100)(51 62 81)(52 63 82)(53 64 83)(54 65 84)(55 66 85)(56 67 86)(57 68 87)(58 69 88)(59 70 89)(60 71 90)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 120)(22 109)(23 118)(24 107)(25 116)(26 105)(27 114)(28 103)(29 112)(30 101)(31 110)(32 119)(33 108)(34 117)(35 106)(36 115)(37 104)(38 113)(39 102)(40 111)(41 89)(42 98)(43 87)(44 96)(45 85)(46 94)(47 83)(48 92)(49 81)(50 90)(51 99)(52 88)(53 97)(54 86)(55 95)(56 84)(57 93)(58 82)(59 91)(60 100)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D4⋊2S3 S3×D5 Q8⋊2D5 C2×S3×D5 D20⋊S3 kernel D20⋊S3 D5×Dic3 C3⋊D20 C3×D20 C5×Dic6 C4×D15 D20 Dic6 C20 D10 C15 Dic3 C12 C5 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 2 1 2 2 4 2 1 2 2 2 4

Matrix representation of D20⋊S3 in GL6(𝔽61)

 0 1 0 0 0 0 60 17 0 0 0 0 0 0 50 0 0 0 0 0 50 11 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 59 0 0 0 0 0 60 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 1 41 0 0 0 0 52 59
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 1 0 0 0 0 0 1 60 0 0 0 0 0 0 60 20 0 0 0 0 0 1

`G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,17,0,0,0,0,0,0,50,50,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,20,1] >;`

D20⋊S3 in GAP, Magma, Sage, TeX

`D_{20}\rtimes S_3`
`% in TeX`

`G:=Group("D20:S3");`
`// GroupNames label`

`G:=SmallGroup(240,127);`
`// by ID`

`G=gap.SmallGroup(240,127);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

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