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G = S3×C50order 300 = 22·3·52

Direct product of C50 and S3

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

Aliases: S3×C50, C6⋊C50, C1503C2, C754C22, C30.4C10, C3⋊(C2×C50), C5.(S3×C10), (C5×S3).C10, (S3×C10).C5, C15.(C2×C10), C10.3(C5×S3), SmallGroup(300,10)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C50
C1C3C15C75S3×C25 — S3×C50
C3 — S3×C50
C1C50

Generators and relations for S3×C50
 G = < a,b,c | a50=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C10
3C10
3C2×C10
3C50
3C50
3C2×C50

Smallest permutation representation of S3×C50
On 150 points
Generators in S150
(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)
(1 114 69)(2 115 70)(3 116 71)(4 117 72)(5 118 73)(6 119 74)(7 120 75)(8 121 76)(9 122 77)(10 123 78)(11 124 79)(12 125 80)(13 126 81)(14 127 82)(15 128 83)(16 129 84)(17 130 85)(18 131 86)(19 132 87)(20 133 88)(21 134 89)(22 135 90)(23 136 91)(24 137 92)(25 138 93)(26 139 94)(27 140 95)(28 141 96)(29 142 97)(30 143 98)(31 144 99)(32 145 100)(33 146 51)(34 147 52)(35 148 53)(36 149 54)(37 150 55)(38 101 56)(39 102 57)(40 103 58)(41 104 59)(42 105 60)(43 106 61)(44 107 62)(45 108 63)(46 109 64)(47 110 65)(48 111 66)(49 112 67)(50 113 68)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)

G:=sub<Sym(150)| (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150), (1,114,69)(2,115,70)(3,116,71)(4,117,72)(5,118,73)(6,119,74)(7,120,75)(8,121,76)(9,122,77)(10,123,78)(11,124,79)(12,125,80)(13,126,81)(14,127,82)(15,128,83)(16,129,84)(17,130,85)(18,131,86)(19,132,87)(20,133,88)(21,134,89)(22,135,90)(23,136,91)(24,137,92)(25,138,93)(26,139,94)(27,140,95)(28,141,96)(29,142,97)(30,143,98)(31,144,99)(32,145,100)(33,146,51)(34,147,52)(35,148,53)(36,149,54)(37,150,55)(38,101,56)(39,102,57)(40,103,58)(41,104,59)(42,105,60)(43,106,61)(44,107,62)(45,108,63)(46,109,64)(47,110,65)(48,111,66)(49,112,67)(50,113,68), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)>;

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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150), (1,114,69)(2,115,70)(3,116,71)(4,117,72)(5,118,73)(6,119,74)(7,120,75)(8,121,76)(9,122,77)(10,123,78)(11,124,79)(12,125,80)(13,126,81)(14,127,82)(15,128,83)(16,129,84)(17,130,85)(18,131,86)(19,132,87)(20,133,88)(21,134,89)(22,135,90)(23,136,91)(24,137,92)(25,138,93)(26,139,94)(27,140,95)(28,141,96)(29,142,97)(30,143,98)(31,144,99)(32,145,100)(33,146,51)(34,147,52)(35,148,53)(36,149,54)(37,150,55)(38,101,56)(39,102,57)(40,103,58)(41,104,59)(42,105,60)(43,106,61)(44,107,62)(45,108,63)(46,109,64)(47,110,65)(48,111,66)(49,112,67)(50,113,68), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,135)(66,136)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120) );

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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)], [(1,114,69),(2,115,70),(3,116,71),(4,117,72),(5,118,73),(6,119,74),(7,120,75),(8,121,76),(9,122,77),(10,123,78),(11,124,79),(12,125,80),(13,126,81),(14,127,82),(15,128,83),(16,129,84),(17,130,85),(18,131,86),(19,132,87),(20,133,88),(21,134,89),(22,135,90),(23,136,91),(24,137,92),(25,138,93),(26,139,94),(27,140,95),(28,141,96),(29,142,97),(30,143,98),(31,144,99),(32,145,100),(33,146,51),(34,147,52),(35,148,53),(36,149,54),(37,150,55),(38,101,56),(39,102,57),(40,103,58),(41,104,59),(42,105,60),(43,106,61),(44,107,62),(45,108,63),(46,109,64),(47,110,65),(48,111,66),(49,112,67),(50,113,68)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,127),(58,128),(59,129),(60,130),(61,131),(62,132),(63,133),(64,134),(65,135),(66,136),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120)])

150 conjugacy classes

class 1 2A2B2C 3 5A5B5C5D 6 10A10B10C10D10E···10L15A15B15C15D25A···25T30A30B30C30D50A···50T50U···50BH75A···75T150A···150T
order12223555561010101010···101515151525···253030303050···5050···5075···75150···150
size113321111211113···322221···122221···13···32···22···2

150 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C5C10C10C25C50C50S3D6C5×S3S3×C10S3×C25S3×C50
kernelS3×C50S3×C25C150S3×C10C5×S3C30D6S3C6C50C25C10C5C2C1
# reps12148420402011442020

Matrix representation of S3×C50 in GL2(𝔽151) generated by

790
079
,
0150
1150
,
0150
1500
G:=sub<GL(2,GF(151))| [79,0,0,79],[0,1,150,150],[0,150,150,0] >;

S3×C50 in GAP, Magma, Sage, TeX

S_3\times C_{50}
% in TeX

G:=Group("S3xC50");
// GroupNames label

G:=SmallGroup(300,10);
// by ID

G=gap.SmallGroup(300,10);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-3,87,5004]);
// Polycyclic

G:=Group<a,b,c|a^50=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×C50 in TeX

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