direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C75, C3⋊C150, C75⋊3C6, C32⋊1C50, C15.2C30, (C3×C75)⋊4C2, C5.(S3×C15), (S3×C15).C5, (C5×S3).C15, C15.5(C5×S3), (C3×C15).1C10, SmallGroup(450,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C75 |
Generators and relations for S3×C75
G = < a,b,c | a75=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C75
►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 26 51)(2 27 52)(3 28 53)(4 29 54)(5 30 55)(6 31 56)(7 32 57)(8 33 58)(9 34 59)(10 35 60)(11 36 61)(12 37 62)(13 38 63)(14 39 64)(15 40 65)(16 41 66)(17 42 67)(18 43 68)(19 44 69)(20 45 70)(21 46 71)(22 47 72)(23 48 73)(24 49 74)(25 50 75)(76 126 101)(77 127 102)(78 128 103)(79 129 104)(80 130 105)(81 131 106)(82 132 107)(83 133 108)(84 134 109)(85 135 110)(86 136 111)(87 137 112)(88 138 113)(89 139 114)(90 140 115)(91 141 116)(92 142 117)(93 143 118)(94 144 119)(95 145 120)(96 146 121)(97 147 122)(98 148 123)(99 149 124)(100 150 125)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)
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), (1,26,51)(2,27,52)(3,28,53)(4,29,54)(5,30,55)(6,31,56)(7,32,57)(8,33,58)(9,34,59)(10,35,60)(11,36,61)(12,37,62)(13,38,63)(14,39,64)(15,40,65)(16,41,66)(17,42,67)(18,43,68)(19,44,69)(20,45,70)(21,46,71)(22,47,72)(23,48,73)(24,49,74)(25,50,75)(76,126,101)(77,127,102)(78,128,103)(79,129,104)(80,130,105)(81,131,106)(82,132,107)(83,133,108)(84,134,109)(85,135,110)(86,136,111)(87,137,112)(88,138,113)(89,139,114)(90,140,115)(91,141,116)(92,142,117)(93,143,118)(94,144,119)(95,145,120)(96,146,121)(97,147,122)(98,148,123)(99,149,124)(100,150,125), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)>;
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), (1,26,51)(2,27,52)(3,28,53)(4,29,54)(5,30,55)(6,31,56)(7,32,57)(8,33,58)(9,34,59)(10,35,60)(11,36,61)(12,37,62)(13,38,63)(14,39,64)(15,40,65)(16,41,66)(17,42,67)(18,43,68)(19,44,69)(20,45,70)(21,46,71)(22,47,72)(23,48,73)(24,49,74)(25,50,75)(76,126,101)(77,127,102)(78,128,103)(79,129,104)(80,130,105)(81,131,106)(82,132,107)(83,133,108)(84,134,109)(85,135,110)(86,136,111)(87,137,112)(88,138,113)(89,139,114)(90,140,115)(91,141,116)(92,142,117)(93,143,118)(94,144,119)(95,145,120)(96,146,121)(97,147,122)(98,148,123)(99,149,124)(100,150,125), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(36,126)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90) );
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)], [(1,26,51),(2,27,52),(3,28,53),(4,29,54),(5,30,55),(6,31,56),(7,32,57),(8,33,58),(9,34,59),(10,35,60),(11,36,61),(12,37,62),(13,38,63),(14,39,64),(15,40,65),(16,41,66),(17,42,67),(18,43,68),(19,44,69),(20,45,70),(21,46,71),(22,47,72),(23,48,73),(24,49,74),(25,50,75),(76,126,101),(77,127,102),(78,128,103),(79,129,104),(80,130,105),(81,131,106),(82,132,107),(83,133,108),(84,134,109),(85,135,110),(86,136,111),(87,137,112),(88,138,113),(89,139,114),(90,140,115),(91,141,116),(92,142,117),(93,143,118),(94,144,119),(95,145,120),(96,146,121),(97,147,122),(98,148,123),(99,149,124),(100,150,125)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,121),(32,122),(33,123),(34,124),(35,125),(36,126),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,133),(44,134),(45,135),(46,136),(47,137),(48,138),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90)]])
225 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15T | 25A | ··· | 25T | 30A | ··· | 30H | 50A | ··· | 50T | 75A | ··· | 75AN | 75AO | ··· | 75CV | 150A | ··· | 150AN |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 25 | ··· | 25 | 30 | ··· | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C25 | C30 | C50 | C75 | C150 | S3 | C3×S3 | C5×S3 | S3×C15 | S3×C25 | S3×C75 |
| kernel | S3×C75 | C3×C75 | S3×C25 | S3×C15 | C75 | C3×C15 | C5×S3 | C3×S3 | C15 | C32 | S3 | C3 | C75 | C25 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 20 | 8 | 20 | 40 | 40 | 1 | 2 | 4 | 8 | 20 | 40 |
Matrix representation of S3×C75 ►in GL2(𝔽151) generated by
| 116 | 0 |
| 0 | 116 |
| 32 | 19 |
| 0 | 118 |
| 128 | 95 |
| 31 | 23 |
G:=sub<GL(2,GF(151))| [116,0,0,116],[32,0,19,118],[128,31,95,23] >;
S3×C75 in GAP, Magma, Sage, TeX
S_3\times C_{75} % in TeX
G:=Group("S3xC75"); // GroupNames label
G:=SmallGroup(450,6);
// by ID
G=gap.SmallGroup(450,6);
# by ID
G:=PCGroup([5,-2,-3,-5,-5,-3,87,7504]);
// Polycyclic
G:=Group<a,b,c|a^75=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export