direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C52, D6.C26, C12⋊2C26, C156⋊6C2, C26.14D6, Dic3⋊2C26, C78.19C22, C3⋊1(C2×C52), C39⋊9(C2×C4), C2.1(S3×C26), C6.2(C2×C26), (S3×C26).2C2, (Dic3×C13)⋊5C2, SmallGroup(312,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C52 |
Generators and relations for S3×C52
G = < a,b,c | a52=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C52
►On 156 points
Generators in S156
(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)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 107 61)(2 108 62)(3 109 63)(4 110 64)(5 111 65)(6 112 66)(7 113 67)(8 114 68)(9 115 69)(10 116 70)(11 117 71)(12 118 72)(13 119 73)(14 120 74)(15 121 75)(16 122 76)(17 123 77)(18 124 78)(19 125 79)(20 126 80)(21 127 81)(22 128 82)(23 129 83)(24 130 84)(25 131 85)(26 132 86)(27 133 87)(28 134 88)(29 135 89)(30 136 90)(31 137 91)(32 138 92)(33 139 93)(34 140 94)(35 141 95)(36 142 96)(37 143 97)(38 144 98)(39 145 99)(40 146 100)(41 147 101)(42 148 102)(43 149 103)(44 150 104)(45 151 53)(46 152 54)(47 153 55)(48 154 56)(49 155 57)(50 156 58)(51 105 59)(52 106 60)
(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)
G:=sub<Sym(156)| (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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,107,61)(2,108,62)(3,109,63)(4,110,64)(5,111,65)(6,112,66)(7,113,67)(8,114,68)(9,115,69)(10,116,70)(11,117,71)(12,118,72)(13,119,73)(14,120,74)(15,121,75)(16,122,76)(17,123,77)(18,124,78)(19,125,79)(20,126,80)(21,127,81)(22,128,82)(23,129,83)(24,130,84)(25,131,85)(26,132,86)(27,133,87)(28,134,88)(29,135,89)(30,136,90)(31,137,91)(32,138,92)(33,139,93)(34,140,94)(35,141,95)(36,142,96)(37,143,97)(38,144,98)(39,145,99)(40,146,100)(41,147,101)(42,148,102)(43,149,103)(44,150,104)(45,151,53)(46,152,54)(47,153,55)(48,154,56)(49,155,57)(50,156,58)(51,105,59)(52,106,60), (53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)>;
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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,107,61)(2,108,62)(3,109,63)(4,110,64)(5,111,65)(6,112,66)(7,113,67)(8,114,68)(9,115,69)(10,116,70)(11,117,71)(12,118,72)(13,119,73)(14,120,74)(15,121,75)(16,122,76)(17,123,77)(18,124,78)(19,125,79)(20,126,80)(21,127,81)(22,128,82)(23,129,83)(24,130,84)(25,131,85)(26,132,86)(27,133,87)(28,134,88)(29,135,89)(30,136,90)(31,137,91)(32,138,92)(33,139,93)(34,140,94)(35,141,95)(36,142,96)(37,143,97)(38,144,98)(39,145,99)(40,146,100)(41,147,101)(42,148,102)(43,149,103)(44,150,104)(45,151,53)(46,152,54)(47,153,55)(48,154,56)(49,155,57)(50,156,58)(51,105,59)(52,106,60), (53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150) );
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),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,107,61),(2,108,62),(3,109,63),(4,110,64),(5,111,65),(6,112,66),(7,113,67),(8,114,68),(9,115,69),(10,116,70),(11,117,71),(12,118,72),(13,119,73),(14,120,74),(15,121,75),(16,122,76),(17,123,77),(18,124,78),(19,125,79),(20,126,80),(21,127,81),(22,128,82),(23,129,83),(24,130,84),(25,131,85),(26,132,86),(27,133,87),(28,134,88),(29,135,89),(30,136,90),(31,137,91),(32,138,92),(33,139,93),(34,140,94),(35,141,95),(36,142,96),(37,143,97),(38,144,98),(39,145,99),(40,146,100),(41,147,101),(42,148,102),(43,149,103),(44,150,104),(45,151,53),(46,152,54),(47,153,55),(48,154,56),(49,155,57),(50,156,58),(51,105,59),(52,106,60)], [(53,151),(54,152),(55,153),(56,154),(57,155),(58,156),(59,105),(60,106),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 39A | ··· | 39L | 52A | ··· | 52X | 52Y | ··· | 52AV | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C13 | C26 | C26 | C26 | C52 | S3 | D6 | C4×S3 | S3×C13 | S3×C26 | S3×C52 |
| kernel | S3×C52 | Dic3×C13 | C156 | S3×C26 | S3×C13 | C4×S3 | Dic3 | C12 | D6 | S3 | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 12 | 12 | 48 | 1 | 1 | 2 | 12 | 12 | 24 |
Matrix representation of S3×C52 ►in GL2(𝔽157) generated by
| 32 | 0 |
| 0 | 32 |
| 0 | 156 |
| 1 | 156 |
| 1 | 156 |
| 0 | 156 |
G:=sub<GL(2,GF(157))| [32,0,0,32],[0,1,156,156],[1,0,156,156] >;
S3×C52 in GAP, Magma, Sage, TeX
S_3\times C_{52} % in TeX
G:=Group("S3xC52"); // GroupNames label
G:=SmallGroup(312,33);
// by ID
G=gap.SmallGroup(312,33);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-3,266,5204]);
// Polycyclic
G:=Group<a,b,c|a^52=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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