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G = S3×D40order 480 = 25·3·5

Direct product of S3 and D40

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×D40, D207D6, C4022D6, C244D10, D1204C2, C1202C22, D6.11D20, D6016C22, C60.92C23, Dic3.2D20, C51(S3×D8), C87(S3×D5), C31(C2×D40), C152(C2×D8), C3⋊C824D10, (C5×S3)⋊1D8, (S3×C8)⋊1D5, (S3×C40)⋊1C2, (S3×D20)⋊8C2, (C3×D40)⋊2C2, C3⋊D409C2, C10.2(S3×D4), C6.2(C2×D20), C2.7(S3×D20), C30.6(C2×D4), (S3×C10).18D4, (C4×S3).37D10, (C3×D20)⋊14C22, (C5×Dic3).21D4, C12.65(C22×D5), (S3×C20).43C22, C20.142(C22×S3), C4.91(C2×S3×D5), (C5×C3⋊C8)⋊28C22, SmallGroup(480,328)

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×D40
 G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1308 in 152 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×2], S3 [×2], C6, C6 [×2], C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C15, C2×C8, D8 [×4], C2×D4 [×2], C20, C20, D10 [×8], C2×C10, C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C2×D8, C40, C40, D20 [×2], D20 [×4], C2×C20, C22×D5 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], C5×Dic3, C60, S3×D5 [×4], C6×D5 [×2], S3×C10, D30 [×2], D40, D40 [×3], C2×C40, C2×D20 [×2], S3×D8, C5×C3⋊C8, C120, C3⋊D20 [×2], C3×D20 [×2], S3×C20, D60 [×2], C2×S3×D5 [×2], C2×D40, C3⋊D40 [×2], C3×D40, S3×C40, D120, S3×D20 [×2], S3×D40
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C22×S3, C2×D8, D20 [×2], C22×D5, S3×D4, S3×D5, D40 [×2], C2×D20, S3×D8, C2×S3×D5, C2×D40, S3×D20, S3×D40

Smallest permutation representation of S3×D40
On 120 points
Generators in S120
(1 85 67)(2 86 68)(3 87 69)(4 88 70)(5 89 71)(6 90 72)(7 91 73)(8 92 74)(9 93 75)(10 94 76)(11 95 77)(12 96 78)(13 97 79)(14 98 80)(15 99 41)(16 100 42)(17 101 43)(18 102 44)(19 103 45)(20 104 46)(21 105 47)(22 106 48)(23 107 49)(24 108 50)(25 109 51)(26 110 52)(27 111 53)(28 112 54)(29 113 55)(30 114 56)(31 115 57)(32 116 58)(33 117 59)(34 118 60)(35 119 61)(36 120 62)(37 81 63)(38 82 64)(39 83 65)(40 84 66)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 119)(42 120)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
(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 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)

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

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

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

69 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F 12 15A15B20A20B20C20D20E20F20G20H24A24B30A30B40A···40H40I···40P60A60B60C60D120A···120H
order1222222234455666888810101010101012151520202020202020202424303040···4040···4060606060120···120
size113320206060226222404022662266664442222666644442···26···644444···4

69 irreducible representations

dim1111112222222222222444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D8D10D10D10D20D20D40S3×D4S3×D5S3×D8C2×S3×D5S3×D20S3×D40
kernelS3×D40C3⋊D40C3×D40S3×C40D120S3×D20D40C5×Dic3S3×C10S3×C8C40D20C5×S3C3⋊C8C24C4×S3Dic3D6S3C10C8C5C4C2C1
# reps12111211121242224416122248

Matrix representation of S3×D40 in GL6(𝔽241)

100000
010000
001000
000100
00000240
00001240
,
100000
010000
00240000
00024000
000001
000010
,
1971630000
78780000
001957000
002082400
00002400
00000240
,
100000
1902400000
004714700
004419400
000010
000001

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[197,78,0,0,0,0,163,78,0,0,0,0,0,0,195,208,0,0,0,0,70,24,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,190,0,0,0,0,0,240,0,0,0,0,0,0,47,44,0,0,0,0,147,194,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×D40 in GAP, Magma, Sage, TeX

S_3\times D_{40}
% in TeX

G:=Group("S3xD40");
// GroupNames label

G:=SmallGroup(480,328);
// by ID

G=gap.SmallGroup(480,328);
# by ID

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

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

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