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G = S3×C40order 240 = 24·3·5

Direct product of C40 and S3

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

Aliases: S3×C40, C244C10, C12012C2, D6.2C20, C20.56D6, C60.73C22, Dic3.2C20, C3⋊C86C10, C31(C2×C40), C1512(C2×C8), C2.1(S3×C20), C6.1(C2×C20), C40(C5×Dic3), (C4×S3).3C10, (S3×C20).6C2, (S3×C10).6C4, C10.22(C4×S3), C4.12(S3×C10), C30.45(C2×C4), C12.12(C2×C10), (C5×Dic3).6C4, C40(C5×C3⋊C8), (C5×C3⋊C8)⋊13C2, SmallGroup(240,49)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C40
C1C3C6C12C60S3×C20 — S3×C40
C3 — S3×C40
C1C40

Generators and relations for S3×C40
 G = < a,b,c | a40=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C4
3C10
3C10
3C8
3C2×C4
3C20
3C2×C10
3C2×C8
3C2×C20
3C40
3C2×C40

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

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

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

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

S3×C40 is a maximal subgroup of   C40.52D6  C40.54D6  C40.55D6  D6.1D20  D407S3  D1205C2

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++++++
imageC1C2C2C2C4C4C5C8C10C10C10C20C20C40S3D6C4×S3C5×S3S3×C8S3×C10S3×C20S3×C40
kernelS3×C40C5×C3⋊C8C120S3×C20C5×Dic3S3×C10S3×C8C5×S3C3⋊C8C24C4×S3Dic3D6S3C40C20C10C8C5C4C2C1
# reps111122484448832112444816

Matrix representation of S3×C40 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] >;

S3×C40 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 S3×C40 in TeX

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