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G = S3×C40⋊C2order 480 = 25·3·5

Direct product of S3 and C40⋊C2

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

Aliases: S3×C40⋊C2, C4023D6, C2413D10, Dic107D6, D20.18D6, D6.10D20, C12012C22, C60.91C23, Dic3.1D20, D60.25C22, Dic3014C22, C88(S3×D5), C3⋊C823D10, (S3×C8)⋊4D5, C51(S3×SD16), (S3×C40)⋊4C2, C2.6(S3×D20), C30.5(C2×D4), C6.1(C2×D20), C10.1(S3×D4), C152(C2×SD16), (C5×S3)⋊1SD16, (S3×D20).2C2, C24⋊D511C2, (S3×Dic10)⋊8C2, (S3×C10).17D4, (C4×S3).36D10, C6.D209C2, C15⋊SD169C2, (C5×Dic3).20D4, C12.64(C22×D5), (S3×C20).42C22, C20.141(C22×S3), (C3×D20).20C22, (C3×Dic10)⋊12C22, C31(C2×C40⋊C2), C4.90(C2×S3×D5), (C3×C40⋊C2)⋊3C2, (C5×C3⋊C8)⋊27C22, SmallGroup(480,327)

Series: Derived Chief Lower central Upper central

C1C60 — S3×C40⋊C2
C1C5C15C30C60C3×D20S3×D20 — S3×C40⋊C2
C15C30C60 — S3×C40⋊C2
C1C2C4C8

Generators and relations for S3×C40⋊C2
 G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c19 >

Subgroups: 972 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], S3, C6, C6, C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, D10 [×4], C2×C10, C3⋊C8, C24, Dic6 [×2], C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C2×SD16, C40, C40, Dic10, Dic10 [×2], D20, D20 [×2], C2×Dic5, C2×C20, C22×D5, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C40⋊C2, C40⋊C2 [×3], C2×C40, C2×Dic10, C2×D20, S3×SD16, C5×C3⋊C8, C120, S3×Dic5, C3⋊D20, C15⋊Q8, C3×Dic10, C3×D20, S3×C20, Dic30, D60, C2×S3×D5, C2×C40⋊C2, C6.D20, C15⋊SD16, C3×C40⋊C2, S3×C40, C24⋊D5, S3×Dic10, S3×D20, S3×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C22×S3, C2×SD16, D20 [×2], C22×D5, S3×D4, S3×D5, C40⋊C2 [×2], C2×D20, S3×SD16, C2×S3×D5, C2×C40⋊C2, S3×D20, S3×C40⋊C2

Smallest permutation representation of S3×C40⋊C2
On 120 points
Generators in S120
(1 92 75)(2 93 76)(3 94 77)(4 95 78)(5 96 79)(6 97 80)(7 98 41)(8 99 42)(9 100 43)(10 101 44)(11 102 45)(12 103 46)(13 104 47)(14 105 48)(15 106 49)(16 107 50)(17 108 51)(18 109 52)(19 110 53)(20 111 54)(21 112 55)(22 113 56)(23 114 57)(24 115 58)(25 116 59)(26 117 60)(27 118 61)(28 119 62)(29 120 63)(30 81 64)(31 82 65)(32 83 66)(33 84 67)(34 85 68)(35 86 69)(36 87 70)(37 88 71)(38 89 72)(39 90 73)(40 91 74)
(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 69)(42 48)(43 67)(44 46)(45 65)(47 63)(49 61)(50 80)(51 59)(52 78)(53 57)(54 76)(56 74)(58 72)(60 70)(62 68)(64 66)(71 79)(73 77)(81 83)(82 102)(84 100)(85 119)(86 98)(87 117)(88 96)(89 115)(90 94)(91 113)(93 111)(95 109)(97 107)(99 105)(101 103)(104 120)(106 118)(108 116)(110 114)

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

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

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

69 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B8A8B8C8D10A10B10C10D10E10F12A12B15A15B20A20B20C20D20E20F20G20H24A24B30A30B40A···40H40I···40P60A60B60C60D120A···120H
order12222234444556688881010101010101212151520202020202020202424303040···4040···4060606060120···120
size113320602262060222402266226666440442222666644442···26···644444···4

69 irreducible representations

dim1111111122222222222222444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10D20D20C40⋊C2S3×D4S3×D5S3×SD16C2×S3×D5S3×D20S3×C40⋊C2
kernelS3×C40⋊C2C6.D20C15⋊SD16C3×C40⋊C2S3×C40C24⋊D5S3×Dic10S3×D20C40⋊C2C5×Dic3S3×C10S3×C8C40Dic10D20C5×S3C3⋊C8C24C4×S3Dic3D6S3C10C8C5C4C2C1
# reps11111111111211142224416122248

Matrix representation of S3×C40⋊C2 in GL4(𝔽241) generated by

24024000
1000
0010
0001
,
1000
24024000
0010
0001
,
240000
024000
0020031
0021094
,
1000
0100
002400
001901
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,200,210,0,0,31,94],[1,0,0,0,0,1,0,0,0,0,240,190,0,0,0,1] >;

S3×C40⋊C2 in GAP, Magma, Sage, TeX

S_3\times C_{40}\rtimes C_2
% in TeX

G:=Group("S3xC40:C2");
// GroupNames label

G:=SmallGroup(480,327);
// by ID

G=gap.SmallGroup(480,327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,58,675,80,1356,18822]);
// 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^19>;
// generators/relations

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