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## G = D40⋊S3order 480 = 25·3·5

### 2nd semidirect product of D40 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D40⋊S3
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — S3×D20 — D40⋊S3
 Lower central C15 — C30 — C60 — D40⋊S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D40⋊S3
G = < a,b,c,d | a40=b2=c3=d2=1, bab=a-1, ac=ca, dad=a21, bc=cb, dbd=a20b, dcd=c-1 >

Subgroups: 1020 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C40, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C40⋊C2, D40, D40, C5×M4(2), C2×D20, C4○D20, D8⋊S3, C5×C3⋊C8, C120, D5×Dic3, C15⋊D4, C3⋊D20, C3×D20, S3×C20, Dic30, D60, C2×S3×D5, C8⋊D10, C3⋊D40, C6.D20, C3×D40, C5×C8⋊S3, C24⋊D5, D205S3, S3×D20, D40⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, D20, C22×D5, S3×D4, S3×D5, C2×D20, D8⋊S3, C2×S3×D5, C8⋊D10, S3×D20, D40⋊S3

Smallest permutation representation of D40⋊S3
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(81 82)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)
(1 49 102)(2 50 103)(3 51 104)(4 52 105)(5 53 106)(6 54 107)(7 55 108)(8 56 109)(9 57 110)(10 58 111)(11 59 112)(12 60 113)(13 61 114)(14 62 115)(15 63 116)(16 64 117)(17 65 118)(18 66 119)(19 67 120)(20 68 81)(21 69 82)(22 70 83)(23 71 84)(24 72 85)(25 73 86)(26 74 87)(27 75 88)(28 76 89)(29 77 90)(30 78 91)(31 79 92)(32 80 93)(33 41 94)(34 42 95)(35 43 96)(36 44 97)(37 45 98)(38 46 99)(39 47 100)(40 48 101)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(41 94)(42 115)(43 96)(44 117)(45 98)(46 119)(47 100)(48 81)(49 102)(50 83)(51 104)(52 85)(53 106)(54 87)(55 108)(56 89)(57 110)(58 91)(59 112)(60 93)(61 114)(62 95)(63 116)(64 97)(65 118)(66 99)(67 120)(68 101)(69 82)(70 103)(71 84)(72 105)(73 86)(74 107)(75 88)(76 109)(77 90)(78 111)(79 92)(80 113)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 6A 6B 6C 8A 8B 10A 10B 10C 10D 12 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 30A 30B 40A 40B 40C 40D 40E 40F 40G 40H 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 3 4 4 4 5 5 6 6 6 8 8 10 10 10 10 12 15 15 20 20 20 20 20 20 24 24 30 30 40 40 40 40 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 6 20 20 60 2 2 6 60 2 2 2 40 40 4 12 2 2 12 12 4 4 4 2 2 2 2 12 12 4 4 4 4 4 4 4 4 12 12 12 12 4 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D10 D20 D20 C8⋊C22 S3×D4 S3×D5 D8⋊S3 C2×S3×D5 C8⋊D10 S3×D20 D40⋊S3 kernel D40⋊S3 C3⋊D40 C6.D20 C3×D40 C5×C8⋊S3 C24⋊D5 D20⋊5S3 S3×D20 D40 C5×Dic3 S3×C10 C8⋊S3 C40 D20 C3⋊C8 C24 C4×S3 Dic3 D6 C15 C10 C8 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 4 4 1 1 2 2 2 4 4 8

Matrix representation of D40⋊S3 in GL4(𝔽241) generated by

 16 203 32 165 86 102 172 204 209 76 225 38 69 37 155 139
,
 84 148 168 55 18 157 36 73 73 186 157 93 205 168 223 84
,
 240 0 240 0 0 240 0 240 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 240 0 240 0 0 240 0 240
`G:=sub<GL(4,GF(241))| [16,86,209,69,203,102,76,37,32,172,225,155,165,204,38,139],[84,18,73,205,148,157,186,168,168,36,157,223,55,73,93,84],[240,0,1,0,0,240,0,1,240,0,0,0,0,240,0,0],[1,0,240,0,0,1,0,240,0,0,240,0,0,0,0,240] >;`

D40⋊S3 in GAP, Magma, Sage, TeX

`D_{40}\rtimes S_3`
`% in TeX`

`G:=Group("D40:S3");`
`// GroupNames label`

`G:=SmallGroup(480,330);`
`// by ID`

`G=gap.SmallGroup(480,330);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^21,b*c=c*b,d*b*d=a^20*b,d*c*d=c^-1>;`
`// generators/relations`

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