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## G = D20⋊D6order 480 = 25·3·5

### 6th semidirect product of D20 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20⋊D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — D20⋊D6
 Lower central C15 — C30 — C60 — D20⋊D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D20⋊D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a3b, dcd=c-1 >

Subgroups: 924 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, C12, C12 [×2], D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C3⋊C8, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C22×S3, C5×S3, C3×D5 [×2], D15, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], Q82S3, Q82S3, C2×D12, C3×C4○D4, C3×Dic5, C60, C60, S3×D5 [×2], C6×D5, C6×D5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D4⋊D6, C5×C3⋊C8, C153C8, C5⋊D12, D5×C12, D5×C12, C3×D20, C3×D20, C5×D12, D60, Q8×C15, C2×S3×D5, D40⋊C2, C20.32D6, C15⋊D8, C3⋊D40, C5×Q82S3, Q82D15, D5×D12, C3×Q82D5, D20⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D4⋊D6, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20⋊D6

Smallest permutation representation of D20⋊D6
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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(41 84)(42 83)(43 82)(44 81)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 120)(78 119)(79 118)(80 117)
(1 81 65)(2 90 66 10 82 74)(3 99 67 19 83 63)(4 88 68 8 84 72)(5 97 69 17 85 61)(6 86 70)(7 95 71 15 87 79)(9 93 73 13 89 77)(11 91 75)(12 100 76 20 92 64)(14 98 78 18 94 62)(16 96 80)(21 56 102 23 54 104)(22 45 103 32 55 113)(24 43 105 30 57 111)(25 52 106 39 58 120)(26 41 107 28 59 109)(27 50 108 37 60 118)(29 48 110 35 42 116)(31 46 112 33 44 114)(34 53 115 40 47 101)(36 51 117 38 49 119)
(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)(21 104)(22 103)(23 102)(24 101)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 30A 30B 40A 40B 40C 40D 60A ··· 60F order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 8 8 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 30 30 40 40 40 40 60 ··· 60 size 1 1 10 12 20 60 2 2 4 10 2 2 2 20 20 20 12 60 2 2 24 24 4 4 4 10 10 4 4 4 4 8 8 4 4 12 12 12 12 8 ··· 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C8⋊C22 S3×D5 D4×D5 D4⋊D6 C2×S3×D5 D40⋊C2 D5×C3⋊D4 D20⋊D6 kernel D20⋊D6 C20.32D6 C15⋊D8 C3⋊D40 C5×Q8⋊2S3 Q8⋊2D15 D5×D12 C3×Q8⋊2D5 Q8⋊2D5 C3×Dic5 C6×D5 Q8⋊2S3 C4×D5 D20 C5×Q8 C3⋊C8 D12 C3×Q8 Dic5 D10 C15 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of D20⋊D6 in GL8(𝔽241)

 240 1 0 0 0 0 0 0 50 190 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 192 0 0 0 0 0 0 123 240 0 0 0 0 0 0 118 1 0 240 0 0 0 0 0 1 1 0
,
 1 0 0 0 0 0 0 0 191 240 0 0 0 0 0 0 0 0 171 140 0 0 0 0 0 0 101 70 0 0 0 0 0 0 0 0 240 0 0 192 0 0 0 0 118 0 240 240 0 0 0 0 123 240 0 1 0 0 0 0 0 0 0 1
,
 51 1 0 0 0 0 0 0 51 190 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 118 0 240 0 0 0 0 0 118 0 0 240
,
 190 240 0 0 0 0 0 0 190 51 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 192 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 1 0 0 0 0 0 118 1 0 240

`G:=sub<GL(8,GF(241))| [240,50,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,123,118,0,0,0,0,0,192,240,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0],[1,191,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,171,101,0,0,0,0,0,0,140,70,0,0,0,0,0,0,0,0,240,118,123,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,192,240,1,1],[51,51,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,1,0,118,118,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[190,190,0,0,0,0,0,0,240,51,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,118,0,0,0,0,192,240,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240] >;`

D20⋊D6 in GAP, Magma, Sage, TeX

`D_{20}\rtimes D_6`
`% in TeX`

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

`G:=SmallGroup(480,578);`
`// by ID`

`G=gap.SmallGroup(480,578);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,100,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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