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## G = D20⋊21D6order 480 = 25·3·5

### 4th semidirect product of D20 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20⋊21D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C15⋊D8 — D20⋊21D6
 Lower central C15 — C30 — C60 — D20⋊21D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D2021D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, dbd=a5b, dcd=c-1 >

Subgroups: 620 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5, C10, C10 [×3], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C3⋊C8 [×2], D12 [×2], D12, C2×C12, C2×C12, C3×D4 [×2], C3×Q8, C22×S3, C5×S3 [×2], C3×D5, C30, C30, C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×3], C22×C10, C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C3×C4○D4, C3×Dic5, C60 [×2], C6×D5, S3×C10 [×4], C2×C30, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, D4⋊D6, C153C8 [×2], C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C5×D12 [×2], C5×D12, C2×C60, S3×C2×C10, D4.D10, C15⋊D8 [×2], C20.D6 [×2], C60.7C4, C3×C4○D20, C10×D12, D2021D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, D4⋊D6, C15⋊D4 [×2], C2×S3×D5, D4.D10, C2×C15⋊D4, D2021D6

Smallest permutation representation of D2021D6
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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 80)(17 79)(18 78)(19 77)(20 76)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)
(1 100 106)(2 81 107)(3 82 108)(4 83 109)(5 84 110)(6 85 111)(7 86 112)(8 87 113)(9 88 114)(10 89 115)(11 90 116)(12 91 117)(13 92 118)(14 93 119)(15 94 120)(16 95 101)(17 96 102)(18 97 103)(19 98 104)(20 99 105)(21 71 56 31 61 46)(22 72 57 32 62 47)(23 73 58 33 63 48)(24 74 59 34 64 49)(25 75 60 35 65 50)(26 76 41 36 66 51)(27 77 42 37 67 52)(28 78 43 38 68 53)(29 79 44 39 69 54)(30 80 45 40 70 55)
(1 106)(2 117)(3 108)(4 119)(5 110)(6 101)(7 112)(8 103)(9 114)(10 105)(11 116)(12 107)(13 118)(14 109)(15 120)(16 111)(17 102)(18 113)(19 104)(20 115)(21 76)(22 67)(23 78)(24 69)(25 80)(26 71)(27 62)(28 73)(29 64)(30 75)(31 66)(32 77)(33 68)(34 79)(35 70)(36 61)(37 72)(38 63)(39 74)(40 65)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(81 91)(83 93)(85 95)(87 97)(89 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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,80)(17,79)(18,78)(19,77)(20,76)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90), (1,100,106)(2,81,107)(3,82,108)(4,83,109)(5,84,110)(6,85,111)(7,86,112)(8,87,113)(9,88,114)(10,89,115)(11,90,116)(12,91,117)(13,92,118)(14,93,119)(15,94,120)(16,95,101)(17,96,102)(18,97,103)(19,98,104)(20,99,105)(21,71,56,31,61,46)(22,72,57,32,62,47)(23,73,58,33,63,48)(24,74,59,34,64,49)(25,75,60,35,65,50)(26,76,41,36,66,51)(27,77,42,37,67,52)(28,78,43,38,68,53)(29,79,44,39,69,54)(30,80,45,40,70,55), (1,106)(2,117)(3,108)(4,119)(5,110)(6,101)(7,112)(8,103)(9,114)(10,105)(11,116)(12,107)(13,118)(14,109)(15,120)(16,111)(17,102)(18,113)(19,104)(20,115)(21,76)(22,67)(23,78)(24,69)(25,80)(26,71)(27,62)(28,73)(29,64)(30,75)(31,66)(32,77)(33,68)(34,79)(35,70)(36,61)(37,72)(38,63)(39,74)(40,65)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(81,91)(83,93)(85,95)(87,97)(89,99)>;`

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 C3⋊D4 C3⋊D4 C5⋊D4 C5⋊D4 C8⋊C22 S3×D5 D4⋊D6 C15⋊D4 C2×S3×D5 C15⋊D4 D4.D10 D20⋊21D6 kernel D20⋊21D6 C15⋊D8 C20.D6 C60.7C4 C3×C4○D20 C10×D12 C4○D20 C60 C2×C30 C2×D12 Dic10 D20 C2×C20 D12 C2×C12 C20 C2×C10 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 1 2 2 2 2 2 4 8

Matrix representation of D2021D6 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 91 0 0 0 0 150 0 0 0 0 0 0 0 0 143 0 0 0 0 98 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 143 0 0 0 0 98 0 0 0 0 91 0 0 0 0 150 0 0 0
,
 240 28 0 0 0 0 129 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 2 213 0 0 0 0 112 239 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,0,0,0,0,98,0,0,0,0,143,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,98,0,0,0,0,143,0,0,0],[240,129,0,0,0,0,28,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[2,112,0,0,0,0,213,239,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D2021D6 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{21}D_6`
`% in TeX`

`G:=Group("D20:21D6");`
`// GroupNames label`

`G:=SmallGroup(480,375);`
`// by ID`

`G=gap.SmallGroup(480,375);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,100,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^10*b,d*b*d=a^5*b,d*c*d=c^-1>;`
`// generators/relations`

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