direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C50, C6⋊C50, C150⋊3C2, C75⋊4C22, 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
| C3 — S3×C50 |
Generators and relations for S3×C50
G = < a,b,c | a50=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 97 135)(2 98 136)(3 99 137)(4 100 138)(5 51 139)(6 52 140)(7 53 141)(8 54 142)(9 55 143)(10 56 144)(11 57 145)(12 58 146)(13 59 147)(14 60 148)(15 61 149)(16 62 150)(17 63 101)(18 64 102)(19 65 103)(20 66 104)(21 67 105)(22 68 106)(23 69 107)(24 70 108)(25 71 109)(26 72 110)(27 73 111)(28 74 112)(29 75 113)(30 76 114)(31 77 115)(32 78 116)(33 79 117)(34 80 118)(35 81 119)(36 82 120)(37 83 121)(38 84 122)(39 85 123)(40 86 124)(41 87 125)(42 88 126)(43 89 127)(44 90 128)(45 91 129)(46 92 130)(47 93 131)(48 94 132)(49 95 133)(50 96 134)
(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)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)
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), (1,97,135)(2,98,136)(3,99,137)(4,100,138)(5,51,139)(6,52,140)(7,53,141)(8,54,142)(9,55,143)(10,56,144)(11,57,145)(12,58,146)(13,59,147)(14,60,148)(15,61,149)(16,62,150)(17,63,101)(18,64,102)(19,65,103)(20,66,104)(21,67,105)(22,68,106)(23,69,107)(24,70,108)(25,71,109)(26,72,110)(27,73,111)(28,74,112)(29,75,113)(30,76,114)(31,77,115)(32,78,116)(33,79,117)(34,80,118)(35,81,119)(36,82,120)(37,83,121)(38,84,122)(39,85,123)(40,86,124)(41,87,125)(42,88,126)(43,89,127)(44,90,128)(45,91,129)(46,92,130)(47,93,131)(48,94,132)(49,95,133)(50,96,134), (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)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113)>;
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), (1,97,135)(2,98,136)(3,99,137)(4,100,138)(5,51,139)(6,52,140)(7,53,141)(8,54,142)(9,55,143)(10,56,144)(11,57,145)(12,58,146)(13,59,147)(14,60,148)(15,61,149)(16,62,150)(17,63,101)(18,64,102)(19,65,103)(20,66,104)(21,67,105)(22,68,106)(23,69,107)(24,70,108)(25,71,109)(26,72,110)(27,73,111)(28,74,112)(29,75,113)(30,76,114)(31,77,115)(32,78,116)(33,79,117)(34,80,118)(35,81,119)(36,82,120)(37,83,121)(38,84,122)(39,85,123)(40,86,124)(41,87,125)(42,88,126)(43,89,127)(44,90,128)(45,91,129)(46,92,130)(47,93,131)(48,94,132)(49,95,133)(50,96,134), (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)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,109)(97,110)(98,111)(99,112)(100,113) );
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)], [(1,97,135),(2,98,136),(3,99,137),(4,100,138),(5,51,139),(6,52,140),(7,53,141),(8,54,142),(9,55,143),(10,56,144),(11,57,145),(12,58,146),(13,59,147),(14,60,148),(15,61,149),(16,62,150),(17,63,101),(18,64,102),(19,65,103),(20,66,104),(21,67,105),(22,68,106),(23,69,107),(24,70,108),(25,71,109),(26,72,110),(27,73,111),(28,74,112),(29,75,113),(30,76,114),(31,77,115),(32,78,116),(33,79,117),(34,80,118),(35,81,119),(36,82,120),(37,83,121),(38,84,122),(39,85,123),(40,86,124),(41,87,125),(42,88,126),(43,89,127),(44,90,128),(45,91,129),(46,92,130),(47,93,131),(48,94,132),(49,95,133),(50,96,134)], [(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),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(85,148),(86,149),(87,150),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(96,109),(97,110),(98,111),(99,112),(100,113)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | 15B | 15C | 15D | 25A | ··· | 25T | 30A | 30B | 30C | 30D | 50A | ··· | 50T | 50U | ··· | 50BH | 75A | ··· | 75T | 150A | ··· | 150T |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | C25 | C50 | C50 | S3 | D6 | C5×S3 | S3×C10 | S3×C25 | S3×C50 |
| kernel | S3×C50 | S3×C25 | C150 | S3×C10 | C5×S3 | C30 | D6 | S3 | C6 | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 20 | 40 | 20 | 1 | 1 | 4 | 4 | 20 | 20 |
Matrix representation of S3×C50 ►in GL2(𝔽151) generated by
| 79 | 0 |
| 0 | 79 |
| 0 | 150 |
| 1 | 150 |
| 0 | 150 |
| 150 | 0 |
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
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