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## G = D20⋊5S3order 240 = 24·3·5

### The semidirect product of D20 and S3 acting through Inn(D20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D20⋊5S3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — D20⋊5S3
 Lower central C15 — C30 — D20⋊5S3
 Upper central C1 — C2 — C4

Generators and relations for D205S3
G = < a,b,c,d | a20=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 328 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C3×D5, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, D42S3, C5×Dic3, Dic15, C60, C6×D5, S3×C10, C4○D20, D5×Dic3, C15⋊D4, C3×D20, S3×C20, Dic30, D205S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, D42S3, S3×D5, C4○D20, C2×S3×D5, D205S3

Smallest permutation representation of D205S3
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)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)
(1 47 24)(2 48 25)(3 49 26)(4 50 27)(5 51 28)(6 52 29)(7 53 30)(8 54 31)(9 55 32)(10 56 33)(11 57 34)(12 58 35)(13 59 36)(14 60 37)(15 41 38)(16 42 39)(17 43 40)(18 44 21)(19 45 22)(20 46 23)(61 119 94)(62 120 95)(63 101 96)(64 102 97)(65 103 98)(66 104 99)(67 105 100)(68 106 81)(69 107 82)(70 108 83)(71 109 84)(72 110 85)(73 111 86)(74 112 87)(75 113 88)(76 114 89)(77 115 90)(78 116 91)(79 117 92)(80 118 93)
(1 80)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D10 C4○D20 D4⋊2S3 S3×D5 C2×S3×D5 D20⋊5S3 kernel D20⋊5S3 D5×Dic3 C15⋊D4 C3×D20 S3×C20 Dic30 D20 C4×S3 C20 D10 C15 Dic3 C12 D6 C3 C5 C4 C2 C1 # reps 1 2 2 1 1 1 1 2 1 2 2 2 2 2 8 1 2 2 4

Matrix representation of D205S3 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 4 36 0 0 25 27
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 60 0
,
 60 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 60 0 0 0 1 0 0 0 0 30 44 0 0 17 31
`G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,4,25,0,0,36,27],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,30,17,0,0,44,31] >;`

D205S3 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_5S_3`
`% in TeX`

`G:=Group("D20:5S3");`
`// GroupNames label`

`G:=SmallGroup(240,126);`
`// by ID`

`G=gap.SmallGroup(240,126);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

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