metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊4S3, C24⋊5D7, C8⋊3D21, C168⋊5C2, D42.1C4, C4.13D42, C28.48D6, C21⋊4M4(2), C12.49D14, C84.55C22, Dic21.1C4, C6.6(C4×D7), C21⋊C8⋊4C2, C7⋊2(C8⋊S3), C14.6(C4×S3), C3⋊2(C8⋊D7), C2.3(C4×D21), C42.15(C2×C4), (C4×D21).2C2, SmallGroup(336,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊S3
G = < a,b,c | a56=b3=c2=1, ab=ba, cac=a13, cbc=b-1 >
Smallest permutation representation of C56⋊S3
►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 96 156)(2 97 157)(3 98 158)(4 99 159)(5 100 160)(6 101 161)(7 102 162)(8 103 163)(9 104 164)(10 105 165)(11 106 166)(12 107 167)(13 108 168)(14 109 113)(15 110 114)(16 111 115)(17 112 116)(18 57 117)(19 58 118)(20 59 119)(21 60 120)(22 61 121)(23 62 122)(24 63 123)(25 64 124)(26 65 125)(27 66 126)(28 67 127)(29 68 128)(30 69 129)(31 70 130)(32 71 131)(33 72 132)(34 73 133)(35 74 134)(36 75 135)(37 76 136)(38 77 137)(39 78 138)(40 79 139)(41 80 140)(42 81 141)(43 82 142)(44 83 143)(45 84 144)(46 85 145)(47 86 146)(48 87 147)(49 88 148)(50 89 149)(51 90 150)(52 91 151)(53 92 152)(54 93 153)(55 94 154)(56 95 155)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 153)(58 166)(59 123)(60 136)(61 149)(62 162)(63 119)(64 132)(65 145)(66 158)(67 115)(68 128)(69 141)(70 154)(71 167)(72 124)(73 137)(74 150)(75 163)(76 120)(77 133)(78 146)(79 159)(80 116)(81 129)(82 142)(83 155)(84 168)(85 125)(86 138)(87 151)(88 164)(89 121)(90 134)(91 147)(92 160)(93 117)(94 130)(95 143)(96 156)(97 113)(98 126)(99 139)(100 152)(101 165)(102 122)(103 135)(104 148)(105 161)(106 118)(107 131)(108 144)(109 157)(110 114)(111 127)(112 140)
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), (1,96,156)(2,97,157)(3,98,158)(4,99,159)(5,100,160)(6,101,161)(7,102,162)(8,103,163)(9,104,164)(10,105,165)(11,106,166)(12,107,167)(13,108,168)(14,109,113)(15,110,114)(16,111,115)(17,112,116)(18,57,117)(19,58,118)(20,59,119)(21,60,120)(22,61,121)(23,62,122)(24,63,123)(25,64,124)(26,65,125)(27,66,126)(28,67,127)(29,68,128)(30,69,129)(31,70,130)(32,71,131)(33,72,132)(34,73,133)(35,74,134)(36,75,135)(37,76,136)(38,77,137)(39,78,138)(40,79,139)(41,80,140)(42,81,141)(43,82,142)(44,83,143)(45,84,144)(46,85,145)(47,86,146)(48,87,147)(49,88,148)(50,89,149)(51,90,150)(52,91,151)(53,92,152)(54,93,153)(55,94,154)(56,95,155), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,153)(58,166)(59,123)(60,136)(61,149)(62,162)(63,119)(64,132)(65,145)(66,158)(67,115)(68,128)(69,141)(70,154)(71,167)(72,124)(73,137)(74,150)(75,163)(76,120)(77,133)(78,146)(79,159)(80,116)(81,129)(82,142)(83,155)(84,168)(85,125)(86,138)(87,151)(88,164)(89,121)(90,134)(91,147)(92,160)(93,117)(94,130)(95,143)(96,156)(97,113)(98,126)(99,139)(100,152)(101,165)(102,122)(103,135)(104,148)(105,161)(106,118)(107,131)(108,144)(109,157)(110,114)(111,127)(112,140)>;
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), (1,96,156)(2,97,157)(3,98,158)(4,99,159)(5,100,160)(6,101,161)(7,102,162)(8,103,163)(9,104,164)(10,105,165)(11,106,166)(12,107,167)(13,108,168)(14,109,113)(15,110,114)(16,111,115)(17,112,116)(18,57,117)(19,58,118)(20,59,119)(21,60,120)(22,61,121)(23,62,122)(24,63,123)(25,64,124)(26,65,125)(27,66,126)(28,67,127)(29,68,128)(30,69,129)(31,70,130)(32,71,131)(33,72,132)(34,73,133)(35,74,134)(36,75,135)(37,76,136)(38,77,137)(39,78,138)(40,79,139)(41,80,140)(42,81,141)(43,82,142)(44,83,143)(45,84,144)(46,85,145)(47,86,146)(48,87,147)(49,88,148)(50,89,149)(51,90,150)(52,91,151)(53,92,152)(54,93,153)(55,94,154)(56,95,155), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,153)(58,166)(59,123)(60,136)(61,149)(62,162)(63,119)(64,132)(65,145)(66,158)(67,115)(68,128)(69,141)(70,154)(71,167)(72,124)(73,137)(74,150)(75,163)(76,120)(77,133)(78,146)(79,159)(80,116)(81,129)(82,142)(83,155)(84,168)(85,125)(86,138)(87,151)(88,164)(89,121)(90,134)(91,147)(92,160)(93,117)(94,130)(95,143)(96,156)(97,113)(98,126)(99,139)(100,152)(101,165)(102,122)(103,135)(104,148)(105,161)(106,118)(107,131)(108,144)(109,157)(110,114)(111,127)(112,140) );
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)], [(1,96,156),(2,97,157),(3,98,158),(4,99,159),(5,100,160),(6,101,161),(7,102,162),(8,103,163),(9,104,164),(10,105,165),(11,106,166),(12,107,167),(13,108,168),(14,109,113),(15,110,114),(16,111,115),(17,112,116),(18,57,117),(19,58,118),(20,59,119),(21,60,120),(22,61,121),(23,62,122),(24,63,123),(25,64,124),(26,65,125),(27,66,126),(28,67,127),(29,68,128),(30,69,129),(31,70,130),(32,71,131),(33,72,132),(34,73,133),(35,74,134),(36,75,135),(37,76,136),(38,77,137),(39,78,138),(40,79,139),(41,80,140),(42,81,141),(43,82,142),(44,83,143),(45,84,144),(46,85,145),(47,86,146),(48,87,147),(49,88,148),(50,89,149),(51,90,150),(52,91,151),(53,92,152),(54,93,153),(55,94,154),(56,95,155)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,153),(58,166),(59,123),(60,136),(61,149),(62,162),(63,119),(64,132),(65,145),(66,158),(67,115),(68,128),(69,141),(70,154),(71,167),(72,124),(73,137),(74,150),(75,163),(76,120),(77,133),(78,146),(79,159),(80,116),(81,129),(82,142),(83,155),(84,168),(85,125),(86,138),(87,151),(88,164),(89,121),(90,134),(91,147),(92,160),(93,117),(94,130),(95,143),(96,156),(97,113),(98,126),(99,139),(100,152),(101,165),(102,122),(103,135),(104,148),(105,161),(106,118),(107,131),(108,144),(109,157),(110,114),(111,127),(112,140)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 42 | 2 | 1 | 1 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D7 | M4(2) | C4×S3 | D14 | D21 | C8⋊S3 | C4×D7 | D42 | C8⋊D7 | C4×D21 | C56⋊S3 |
| kernel | C56⋊S3 | C21⋊C8 | C168 | C4×D21 | Dic21 | D42 | C56 | C28 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C56⋊S3 ►in GL2(𝔽337) generated by
| 37 | 13 |
| 324 | 292 |
| 261 | 161 |
| 176 | 75 |
| 1 | 0 |
| 227 | 336 |
G:=sub<GL(2,GF(337))| [37,324,13,292],[261,176,161,75],[1,227,0,336] >;
C56⋊S3 in GAP, Magma, Sage, TeX
C_{56}\rtimes S_3 % in TeX
G:=Group("C56:S3"); // GroupNames label
G:=SmallGroup(336,91);
// by ID
G=gap.SmallGroup(336,91);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,50,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^56=b^3=c^2=1,a*b=b*a,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations
Export