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G = D40⋊S3order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C402D6, D402S3, D208D6, C2410D10, D6.5D20, C1208C22, C60.94C23, Dic3.7D20, D60.26C22, Dic3015C22, C3⋊C82D10, C84(S3×D5), (C3×D40)⋊4C2, C8⋊S33D5, C51(D8⋊S3), C10.4(S3×D4), C30.8(C2×D4), C2.9(S3×D20), C6.4(C2×D20), C24⋊D53C2, (S3×D20)⋊10C2, C3⋊D4011C2, C32(C8⋊D10), C154(C8⋊C22), (C4×S3).2D10, (S3×C10).2D4, D205S38C2, (C5×Dic3).2D4, C6.D2010C2, (C3×D20)⋊15C22, C12.67(C22×D5), (S3×C20).25C22, C20.144(C22×S3), C4.93(C2×S3×D5), (C5×C8⋊S3)⋊3C2, (C5×C3⋊C8)⋊16C22, SmallGroup(480,330)

Series: Derived Chief Lower central Upper central

C1C60 — D40⋊S3
C1C5C15C30C60C3×D20S3×D20 — D40⋊S3
C15C30C60 — D40⋊S3
C1C2C4C8

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 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C5×S3, C3×D5 [×2], D15, C30, C8⋊C22, C40, C40, Dic10, C4×D5, D20 [×2], D20 [×2], 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 [×2], C6×D5 [×2], S3×C10, D30, C40⋊C2 [×2], D40, D40, C5×M4(2), C2×D20, C4○D20, D8⋊S3, C5×C3⋊C8, C120, D5×Dic3, C15⋊D4, C3⋊D20, C3×D20 [×2], 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 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, D20 [×2], 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 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 86)(82 85)(83 84)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)
(1 84 74)(2 85 75)(3 86 76)(4 87 77)(5 88 78)(6 89 79)(7 90 80)(8 91 41)(9 92 42)(10 93 43)(11 94 44)(12 95 45)(13 96 46)(14 97 47)(15 98 48)(16 99 49)(17 100 50)(18 101 51)(19 102 52)(20 103 53)(21 104 54)(22 105 55)(23 106 56)(24 107 57)(25 108 58)(26 109 59)(27 110 60)(28 111 61)(29 112 62)(30 113 63)(31 114 64)(32 115 65)(33 116 66)(34 117 67)(35 118 68)(36 119 69)(37 120 70)(38 81 71)(39 82 72)(40 83 73)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(41 111)(42 92)(43 113)(44 94)(45 115)(46 96)(47 117)(48 98)(49 119)(50 100)(51 81)(52 102)(53 83)(54 104)(55 85)(56 106)(57 87)(58 108)(59 89)(60 110)(61 91)(62 112)(63 93)(64 114)(65 95)(66 116)(67 97)(68 118)(69 99)(70 120)(71 101)(72 82)(73 103)(74 84)(75 105)(76 86)(77 107)(78 88)(79 109)(80 90)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D 12 15A15B20A20B20C20D20E20F24A24B30A30B40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order122222344455666881010101012151520202020202024243030404040404040404060606060120···120
size11620206022660222404041222121244422221212444444441212121244444···4

54 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D10D10D10D20D20C8⋊C22S3×D4S3×D5D8⋊S3C2×S3×D5C8⋊D10S3×D20D40⋊S3
kernelD40⋊S3C3⋊D40C6.D20C3×D40C5×C8⋊S3C24⋊D5D205S3S3×D20D40C5×Dic3S3×C10C8⋊S3C40D20C3⋊C8C24C4×S3Dic3D6C15C10C8C5C4C3C2C1
# reps111111111112122224411222448

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

1620332165
86102172204
2097622538
6937155139
,
8414816855
181573673
7318615793
20516822384
,
24002400
02400240
1000
0100
,
1000
0100
24002400
02400240
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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