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## G = D20⋊24D6order 480 = 25·3·5

### 7th semidirect product of D20 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊24D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — D20⋊24D6
 Lower central C15 — C30 — D20⋊24D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D2024D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >

Subgroups: 1676 in 328 conjugacy classes, 110 normal (52 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×5], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×2], C4×S3 [×2], C4×S3 [×8], D12, D12 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×2], C3×D5 [×2], D15 [×2], D15, C30, C30, C2×C4○D4, Dic10, Dic10 [×2], C4×D5 [×2], C4×D5 [×8], D20, D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12, C4○D12 [×2], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C4×D5 [×3], C4○D20, C4○D20 [×2], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C2×S3×D5 [×2], C22×D15, D5×C4○D4, D20⋊S3, D12⋊D5, D15⋊Q8, D6.D10 [×2], C4×S3×D5 [×2], C20⋊D6, C30.C23 [×2], D10⋊D6 [×2], C3×C4○D20, C5×C4○D12, C2×C4×D15, D2024D6
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, D2024D6

Smallest permutation representation of D2024D6
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 29 67 97 116 52)(2 30 68 98 117 53)(3 31 69 99 118 54)(4 32 70 100 119 55)(5 33 71 81 120 56)(6 34 72 82 101 57)(7 35 73 83 102 58)(8 36 74 84 103 59)(9 37 75 85 104 60)(10 38 76 86 105 41)(11 39 77 87 106 42)(12 40 78 88 107 43)(13 21 79 89 108 44)(14 22 80 90 109 45)(15 23 61 91 110 46)(16 24 62 92 111 47)(17 25 63 93 112 48)(18 26 64 94 113 49)(19 27 65 95 114 50)(20 28 66 96 115 51)
(1 42)(2 51)(3 60)(4 49)(5 58)(6 47)(7 56)(8 45)(9 54)(10 43)(11 52)(12 41)(13 50)(14 59)(15 48)(16 57)(17 46)(18 55)(19 44)(20 53)(21 114)(22 103)(23 112)(24 101)(25 110)(26 119)(27 108)(28 117)(29 106)(30 115)(31 104)(32 113)(33 102)(34 111)(35 120)(36 109)(37 118)(38 107)(39 116)(40 105)(61 93)(62 82)(63 91)(64 100)(65 89)(66 98)(67 87)(68 96)(69 85)(70 94)(71 83)(72 92)(73 81)(74 90)(75 99)(76 88)(77 97)(78 86)(79 95)(80 84)```

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

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

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

66 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 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 30A ··· 30F 60A ··· 60H 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 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 2 6 6 10 10 15 15 30 2 1 1 2 6 6 10 10 15 15 30 2 2 2 4 20 20 2 2 4 4 12 12 12 12 2 2 4 20 20 4 4 2 2 2 2 4 4 12 12 12 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 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 type + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 D10 S3×D5 S3×C4○D4 C2×S3×D5 C2×S3×D5 D5×C4○D4 D20⋊24D6 kernel D20⋊24D6 D20⋊S3 D12⋊D5 D15⋊Q8 D6.D10 C4×S3×D5 C20⋊D6 C30.C23 D10⋊D6 C3×C4○D20 C5×C4○D12 C2×C4×D15 C4○D20 C4○D12 Dic10 C4×D5 D20 C5⋊D4 C2×C20 D15 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C2×C4 C5 C4 C22 C3 C1 # reps 1 1 1 1 2 2 1 2 2 1 1 1 1 2 1 2 1 2 1 4 2 4 2 4 2 2 2 4 2 4 8

Matrix representation of D2024D6 in GL6(𝔽61)

 44 60 0 0 0 0 1 0 0 0 0 0 0 0 0 50 0 0 0 0 50 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 44 0 0 0 0 0 1 0 0 0 0 0 0 0 50 0 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 41 0 0 0 0 52 59
,
 44 17 0 0 0 0 1 17 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 2 41 0 0 0 0 52 59

`G:=sub<GL(6,GF(61))| [44,1,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,44,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[44,1,0,0,0,0,17,17,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,2,52,0,0,0,0,41,59] >;`

D2024D6 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{24}D_6`
`% in TeX`

`G:=Group("D20:24D6");`
`// GroupNames label`

`G:=SmallGroup(480,1092);`
`// by ID`

`G=gap.SmallGroup(480,1092);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,1356,18822]);`
`// 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^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

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