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G = D6⋊4D20order 480 = 25·3·5

1st semidirect product of D6 and D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — D6⋊4D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C22×S3×D5 — D6⋊4D20
 Lower central C15 — C2×C30 — D6⋊4D20
 Upper central C1 — C22 — C2×C4

Generators and relations for D64D20
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, dad=a19, cbc-1=a15b, dbd=a3b, dcd=c-1 >

Subgroups: 1820 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×4], C6 [×3], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], D5 [×5], C10 [×3], C10 [×2], Dic3 [×2], C12, D6 [×2], D6 [×14], C2×C6, C2×C6 [×7], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×7], C22×C6 [×2], C5×S3 [×2], C3×D5 [×3], D15 [×2], C30 [×3], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5 [×2], C22×D5 [×7], C22×C10, D6⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C5×Dic3, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×5], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, D10⋊C4 [×2], C5×C22⋊C4, C2×D20, C2×D20, C2×C5⋊D4, C23×D5, C232D6, C15⋊D4 [×2], C3⋊D20 [×2], C3×D20 [×2], C10×Dic3, C2×Dic15, C2×C60, C2×S3×D5 [×6], D5×C2×C6 [×2], S3×C2×C10, C22×D15, C22⋊D20, D10⋊Dic3, C5×D6⋊C4, D303C4, C2×C15⋊D4, C2×C3⋊D20, C6×D20, C22×S3×D5, D64D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, D20 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, S3×D5, C2×D20, D4×D5 [×2], C232D6, C2×S3×D5, C22⋊D20, S3×D20, C20⋊D6, D5×C3⋊D4, D64D20

Smallest permutation representation of D64D20
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 39)(32 38)(33 37)(34 36)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(110 120)(111 119)(112 118)(113 117)(114 116)
(1 43 89 93)(2 44 90 94)(3 45 61 95)(4 46 62 96)(5 47 63 97)(6 48 64 98)(7 49 65 99)(8 50 66 100)(9 51 67 101)(10 52 68 102)(11 53 69 103)(12 54 70 104)(13 55 71 105)(14 56 72 106)(15 57 73 107)(16 58 74 108)(17 59 75 109)(18 60 76 110)(19 31 77 111)(20 32 78 112)(21 33 79 113)(22 34 80 114)(23 35 81 115)(24 36 82 116)(25 37 83 117)(26 38 84 118)(27 39 85 119)(28 40 86 120)(29 41 87 91)(30 42 88 92)
(1 93)(2 112)(3 101)(4 120)(5 109)(6 98)(7 117)(8 106)(9 95)(10 114)(11 103)(12 92)(13 111)(14 100)(15 119)(16 108)(17 97)(18 116)(19 105)(20 94)(21 113)(22 102)(23 91)(24 110)(25 99)(26 118)(27 107)(28 96)(29 115)(30 104)(31 71)(32 90)(33 79)(34 68)(35 87)(36 76)(37 65)(38 84)(39 73)(40 62)(41 81)(42 70)(43 89)(44 78)(45 67)(46 86)(47 75)(48 64)(49 83)(50 72)(51 61)(52 80)(53 69)(54 88)(55 77)(56 66)(57 85)(58 74)(59 63)(60 82)```

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

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

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

60 conjugacy classes

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

60 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 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D5 D6 D6 D10 D10 D10 C3⋊D4 D20 S3×D4 S3×D5 D4×D5 C2×S3×D5 S3×D20 C20⋊D6 D5×C3⋊D4 kernel D6⋊4D20 D10⋊Dic3 C5×D6⋊C4 D30⋊3C4 C2×C15⋊D4 C2×C3⋊D20 C6×D20 C22×S3×D5 C2×D20 C6×D5 S3×C10 D30 D6⋊C4 C2×C20 C22×D5 C2×Dic3 C2×C12 C22×S3 D10 D6 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 4 8 2 2 4 2 4 4 4

Matrix representation of D64D20 in GL8(𝔽61)

 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 44 60 0 0 0 0 0 0 45 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 53 1 0 0 0 0 0 0 0 0 0 43 0 0 0 0 0 0 44 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 11 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 46 0 0 0 0 0 0 53 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 27 16 0 0 0 0 0 0 46 34
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 46 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 17 1 0 0 0 0 0 0 17 44 0 0 0 0 0 0 0 0 27 16 0 0 0 0 0 0 46 34

`G:=sub<GL(8,GF(61))| [0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,44,45,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,53,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,44,0,0,0,0,0,0,43,0,0,0,0,0,0,0,0,0,60,11,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,53,0,0,0,0,0,0,46,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,16,34],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,46,1,0,0,0,0,0,0,0,0,17,17,0,0,0,0,0,0,1,44,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,16,34] >;`

D64D20 in GAP, Magma, Sage, TeX

`D_6\rtimes_4D_{20}`
`% in TeX`

`G:=Group("D6:4D20");`
`// GroupNames label`

`G:=SmallGroup(480,550);`
`// by ID`

`G=gap.SmallGroup(480,550);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^19,c*b*c^-1=a^15*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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