direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C56, C24⋊4C14, C168⋊12C2, D6.2C28, C28.56D6, C84.73C22, Dic3.2C28, C3⋊C8⋊6C14, C3⋊1(C2×C56), C21⋊6(C2×C8), C2.1(S3×C28), C6.1(C2×C28), (C4×S3).3C14, (S3×C28).6C2, (S3×C14).4C4, C4.12(S3×C14), C14.15(C4×S3), C42.24(C2×C4), C12.12(C2×C14), (C7×Dic3).4C4, (C7×C3⋊C8)⋊13C2, SmallGroup(336,74)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C56 |
Generators and relations for S3×C56
G = < a,b,c | a56=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C56
►On 168 points
Generators in S168
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 75 130)(2 76 131)(3 77 132)(4 78 133)(5 79 134)(6 80 135)(7 81 136)(8 82 137)(9 83 138)(10 84 139)(11 85 140)(12 86 141)(13 87 142)(14 88 143)(15 89 144)(16 90 145)(17 91 146)(18 92 147)(19 93 148)(20 94 149)(21 95 150)(22 96 151)(23 97 152)(24 98 153)(25 99 154)(26 100 155)(27 101 156)(28 102 157)(29 103 158)(30 104 159)(31 105 160)(32 106 161)(33 107 162)(34 108 163)(35 109 164)(36 110 165)(37 111 166)(38 112 167)(39 57 168)(40 58 113)(41 59 114)(42 60 115)(43 61 116)(44 62 117)(45 63 118)(46 64 119)(47 65 120)(48 66 121)(49 67 122)(50 68 123)(51 69 124)(52 70 125)(53 71 126)(54 72 127)(55 73 128)(56 74 129)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 140)(58 141)(59 142)(60 143)(61 144)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 161)(79 162)(80 163)(81 164)(82 165)(83 166)(84 167)(85 168)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)
G:=sub<Sym(168)| (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,75,130)(2,76,131)(3,77,132)(4,78,133)(5,79,134)(6,80,135)(7,81,136)(8,82,137)(9,83,138)(10,84,139)(11,85,140)(12,86,141)(13,87,142)(14,88,143)(15,89,144)(16,90,145)(17,91,146)(18,92,147)(19,93,148)(20,94,149)(21,95,150)(22,96,151)(23,97,152)(24,98,153)(25,99,154)(26,100,155)(27,101,156)(28,102,157)(29,103,158)(30,104,159)(31,105,160)(32,106,161)(33,107,162)(34,108,163)(35,109,164)(36,110,165)(37,111,166)(38,112,167)(39,57,168)(40,58,113)(41,59,114)(42,60,115)(43,61,116)(44,62,117)(45,63,118)(46,64,119)(47,65,120)(48,66,121)(49,67,122)(50,68,123)(51,69,124)(52,70,125)(53,71,126)(54,72,127)(55,73,128)(56,74,129), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,161)(79,162)(80,163)(81,164)(82,165)(83,166)(84,167)(85,168)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)>;
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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,75,130)(2,76,131)(3,77,132)(4,78,133)(5,79,134)(6,80,135)(7,81,136)(8,82,137)(9,83,138)(10,84,139)(11,85,140)(12,86,141)(13,87,142)(14,88,143)(15,89,144)(16,90,145)(17,91,146)(18,92,147)(19,93,148)(20,94,149)(21,95,150)(22,96,151)(23,97,152)(24,98,153)(25,99,154)(26,100,155)(27,101,156)(28,102,157)(29,103,158)(30,104,159)(31,105,160)(32,106,161)(33,107,162)(34,108,163)(35,109,164)(36,110,165)(37,111,166)(38,112,167)(39,57,168)(40,58,113)(41,59,114)(42,60,115)(43,61,116)(44,62,117)(45,63,118)(46,64,119)(47,65,120)(48,66,121)(49,67,122)(50,68,123)(51,69,124)(52,70,125)(53,71,126)(54,72,127)(55,73,128)(56,74,129), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,161)(79,162)(80,163)(81,164)(82,165)(83,166)(84,167)(85,168)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139) );
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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,75,130),(2,76,131),(3,77,132),(4,78,133),(5,79,134),(6,80,135),(7,81,136),(8,82,137),(9,83,138),(10,84,139),(11,85,140),(12,86,141),(13,87,142),(14,88,143),(15,89,144),(16,90,145),(17,91,146),(18,92,147),(19,93,148),(20,94,149),(21,95,150),(22,96,151),(23,97,152),(24,98,153),(25,99,154),(26,100,155),(27,101,156),(28,102,157),(29,103,158),(30,104,159),(31,105,160),(32,106,161),(33,107,162),(34,108,163),(35,109,164),(36,110,165),(37,111,166),(38,112,167),(39,57,168),(40,58,113),(41,59,114),(42,60,115),(43,61,116),(44,62,117),(45,63,118),(46,64,119),(47,65,120),(48,66,121),(49,67,122),(50,68,123),(51,69,124),(52,70,125),(53,71,126),(54,72,127),(55,73,128),(56,74,129)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,140),(58,141),(59,142),(60,143),(61,144),(62,145),(63,146),(64,147),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,159),(77,160),(78,161),(79,162),(80,163),(81,164),(82,165),(83,166),(84,167),(85,168),(86,113),(87,114),(88,115),(89,116),(90,117),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139)]])
168 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28L | 28M | ··· | 28X | 42A | ··· | 42F | 56A | ··· | 56X | 56Y | ··· | 56AV | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
168 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | S3 | D6 | C4×S3 | S3×C7 | S3×C8 | S3×C14 | S3×C28 | S3×C56 |
| kernel | S3×C56 | C7×C3⋊C8 | C168 | S3×C28 | C7×Dic3 | S3×C14 | S3×C8 | S3×C7 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | S3 | C56 | C28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 12 | 12 | 48 | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 |
Matrix representation of S3×C56 ►in GL3(𝔽337) generated by
| 85 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 336 | 336 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(337))| [85,0,0,0,36,0,0,0,36],[1,0,0,0,336,1,0,336,0],[1,0,0,0,0,1,0,1,0] >;
S3×C56 in GAP, Magma, Sage, TeX
S_3\times C_{56} % in TeX
G:=Group("S3xC56"); // GroupNames label
G:=SmallGroup(336,74);
// by ID
G=gap.SmallGroup(336,74);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,175,69,8069]);
// Polycyclic
G:=Group<a,b,c|a^56=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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