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## G = D20.9D6order 480 = 25·3·5

### 9th non-split extension by D20 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20.9D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — C12.28D10 — D20.9D6
 Lower central C15 — C30 — C60 — D20.9D6
 Upper central C1 — C2 — C4 — D4

Generators and relations for D20.9D6
G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a8b, dbd=a3b, dcd=a10c-1 >

Subgroups: 812 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3, C6, C6 [×3], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C3×D5 [×2], D15, C30, 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], D4.S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5 [×3], D30, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D126C22, C5×C3⋊C8, C153C8, D30.C2, C3⋊D20, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, D60, D4×C15, D5×C2×C6, D40⋊C2, C20.32D6, C3⋊D40, C30.D4, C5×D4.S3, D4⋊D15, C12.28D10, C3×D4×D5, D20.9D6
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, D126C22, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20.9D6

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 20C 20D 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 6 6 6 6 8 8 10 10 10 10 12 12 15 15 20 20 20 20 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 10 20 60 2 2 10 12 2 2 2 4 4 10 10 20 20 12 60 2 2 8 8 4 20 4 4 4 4 24 24 4 4 8 8 8 8 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 D12⋊6C22 C2×S3×D5 D40⋊C2 D5×C3⋊D4 D20.9D6 kernel D20.9D6 C20.32D6 C3⋊D40 C30.D4 C5×D4.S3 D4⋊D15 C12.28D10 C3×D4×D5 D4×D5 C3×Dic5 C6×D5 D4.S3 C4×D5 D20 C5×D4 C3⋊C8 Dic6 C3×D4 Dic5 D10 C15 D4 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 D20.9D6 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 51 188 0 0 0 0 52 240 0 0 124 125 51 240 0 0 234 233 1 189
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 52 0 0 0 0 0 1 0 0 0 0 117 118 240 240 0 0 7 59 0 1
,
 240 240 0 0 0 0 1 0 0 0 0 0 0 0 189 52 0 0 0 0 240 52 0 0 0 0 189 0 51 52 0 0 240 52 191 190
,
 112 221 0 0 0 0 109 129 0 0 0 0 0 0 33 33 73 0 0 0 135 208 108 168 0 0 0 0 212 33 0 0 0 0 106 29

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,124,234,0,0,1,0,125,233,0,0,51,52,51,1,0,0,188,240,240,189],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,117,7,0,0,52,1,118,59,0,0,0,0,240,0,0,0,0,0,240,1],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,189,240,189,240,0,0,52,52,0,52,0,0,0,0,51,191,0,0,0,0,52,190],[112,109,0,0,0,0,221,129,0,0,0,0,0,0,33,135,0,0,0,0,33,208,0,0,0,0,73,108,212,106,0,0,0,168,33,29] >;`

D20.9D6 in GAP, Magma, Sage, TeX

`D_{20}._9D_6`
`% in TeX`

`G:=Group("D20.9D6");`
`// GroupNames label`

`G:=SmallGroup(480,567);`
`// by ID`

`G=gap.SmallGroup(480,567);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,346,185,80,1356,18822]);`
`// 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^8*b,d*b*d=a^3*b,d*c*d=a^10*c^-1>;`
`// generators/relations`

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