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G = S3xC40order 240 = 24·3·5

Direct product of C40 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3xC40, C24:4C10, C120:12C2, D6.2C20, C20.56D6, C60.73C22, Dic3.2C20, C3:C8:6C10, C3:1(C2xC40), C15:12(C2xC8), C2.1(S3xC20), C6.1(C2xC20), C40o(C5xDic3), (C4xS3).3C10, (S3xC20).6C2, (S3xC10).6C4, C10.22(C4xS3), C4.12(S3xC10), C30.45(C2xC4), C12.12(C2xC10), (C5xDic3).6C4, C40o(C5xC3:C8), (C5xC3:C8):13C2, SmallGroup(240,49)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC40
C1C3C6C12C60S3xC20 — S3xC40
C3 — S3xC40
C1C40

Generators and relations for S3xC40
 G = < a,b,c | a40=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 72 in 44 conjugacy classes, 30 normal (26 characteristic)
Quotients: C1, C2, C4, C22, C5, S3, C8, C2xC4, C10, D6, C2xC8, C20, C2xC10, C4xS3, C5xS3, C40, C2xC20, S3xC8, S3xC10, C2xC40, S3xC20, S3xC40
3C2
3C2
3C22
3C4
3C10
3C10
3C8
3C2xC4
3C20
3C2xC10
3C2xC8
3C2xC20
3C40
3C2xC40

Smallest permutation representation of S3xC40
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 64 106)(2 65 107)(3 66 108)(4 67 109)(5 68 110)(6 69 111)(7 70 112)(8 71 113)(9 72 114)(10 73 115)(11 74 116)(12 75 117)(13 76 118)(14 77 119)(15 78 120)(16 79 81)(17 80 82)(18 41 83)(19 42 84)(20 43 85)(21 44 86)(22 45 87)(23 46 88)(24 47 89)(25 48 90)(26 49 91)(27 50 92)(28 51 93)(29 52 94)(30 53 95)(31 54 96)(32 55 97)(33 56 98)(34 57 99)(35 58 100)(36 59 101)(37 60 102)(38 61 103)(39 62 104)(40 63 105)
(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)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(79 101)(80 102)

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

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

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

S3xC40 is a maximal subgroup of   C40.52D6  C40.54D6  C40.55D6  D6.1D20  D40:7S3  D120:5C2

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B5C5D 6 8A8B8C8D8E8F8G8H10A10B10C10D10E···10L12A12B15A15B15C15D20A···20H20I···20P24A24B24C24D30A30B30C30D40A···40P40Q···40AF60A···60H120A···120P
order12223444455556888888881010101010···1012121515151520···2020···20242424243030303040···4040···4060···60120···120
size113321133111121111333311113···32222221···13···3222222221···13···32···22···2

120 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C4C4C5C8C10C10C10C20C20C40S3D6C4xS3C5xS3S3xC8S3xC10S3xC20S3xC40
kernelS3xC40C5xC3:C8C120S3xC20C5xDic3S3xC10S3xC8C5xS3C3:C8C24C4xS3Dic3D6S3C40C20C10C8C5C4C2C1
# reps111122484448832112444816

Matrix representation of S3xC40 in GL2(F41) generated by

60
06
,
4034
60
,
4034
01
G:=sub<GL(2,GF(41))| [6,0,0,6],[40,6,34,0],[40,0,34,1] >;

S3xC40 in GAP, Magma, Sage, TeX

S_3\times C_{40}
% in TeX

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

G:=SmallGroup(240,49);
// by ID

G=gap.SmallGroup(240,49);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,127,69,5765]);
// Polycyclic

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

Export

Subgroup lattice of S3xC40 in TeX

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