direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C56⋊C2, C24⋊6D7, C168⋊6C2, C56⋊10C6, C21⋊8SD16, D28.3C6, C42.23D4, C6.13D28, Dic14⋊7C6, C12.52D14, C84.59C22, C8⋊2(C3×D7), C4.8(C6×D7), C7⋊5(C3×SD16), C2.3(C3×D28), C28.31(C2×C6), (C3×D28).3C2, C14.17(C3×D4), (C3×Dic14)⋊7C2, SmallGroup(336,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C56⋊C2
G = < a,b,c | a3=b56=c2=1, ab=ba, ac=ca, cbc=b27 >
Smallest permutation representation of C3×C56⋊C2
►On 168 points
Generators in S168
(1 78 165)(2 79 166)(3 80 167)(4 81 168)(5 82 113)(6 83 114)(7 84 115)(8 85 116)(9 86 117)(10 87 118)(11 88 119)(12 89 120)(13 90 121)(14 91 122)(15 92 123)(16 93 124)(17 94 125)(18 95 126)(19 96 127)(20 97 128)(21 98 129)(22 99 130)(23 100 131)(24 101 132)(25 102 133)(26 103 134)(27 104 135)(28 105 136)(29 106 137)(30 107 138)(31 108 139)(32 109 140)(33 110 141)(34 111 142)(35 112 143)(36 57 144)(37 58 145)(38 59 146)(39 60 147)(40 61 148)(41 62 149)(42 63 150)(43 64 151)(44 65 152)(45 66 153)(46 67 154)(47 68 155)(48 69 156)(49 70 157)(50 71 158)(51 72 159)(52 73 160)(53 74 161)(54 75 162)(55 76 163)(56 77 164)
(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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 71)(58 98)(59 69)(60 96)(61 67)(62 94)(63 65)(64 92)(66 90)(68 88)(70 86)(72 84)(73 111)(74 82)(75 109)(76 80)(77 107)(79 105)(81 103)(83 101)(85 99)(87 97)(89 95)(91 93)(100 112)(102 110)(104 108)(113 161)(114 132)(115 159)(116 130)(117 157)(118 128)(119 155)(120 126)(121 153)(122 124)(123 151)(125 149)(127 147)(129 145)(131 143)(133 141)(134 168)(135 139)(136 166)(138 164)(140 162)(142 160)(144 158)(146 156)(148 154)(150 152)(163 167)
G:=sub<Sym(168)| (1,78,165)(2,79,166)(3,80,167)(4,81,168)(5,82,113)(6,83,114)(7,84,115)(8,85,116)(9,86,117)(10,87,118)(11,88,119)(12,89,120)(13,90,121)(14,91,122)(15,92,123)(16,93,124)(17,94,125)(18,95,126)(19,96,127)(20,97,128)(21,98,129)(22,99,130)(23,100,131)(24,101,132)(25,102,133)(26,103,134)(27,104,135)(28,105,136)(29,106,137)(30,107,138)(31,108,139)(32,109,140)(33,110,141)(34,111,142)(35,112,143)(36,57,144)(37,58,145)(38,59,146)(39,60,147)(40,61,148)(41,62,149)(42,63,150)(43,64,151)(44,65,152)(45,66,153)(46,67,154)(47,68,155)(48,69,156)(49,70,157)(50,71,158)(51,72,159)(52,73,160)(53,74,161)(54,75,162)(55,76,163)(56,77,164), (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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,71)(58,98)(59,69)(60,96)(61,67)(62,94)(63,65)(64,92)(66,90)(68,88)(70,86)(72,84)(73,111)(74,82)(75,109)(76,80)(77,107)(79,105)(81,103)(83,101)(85,99)(87,97)(89,95)(91,93)(100,112)(102,110)(104,108)(113,161)(114,132)(115,159)(116,130)(117,157)(118,128)(119,155)(120,126)(121,153)(122,124)(123,151)(125,149)(127,147)(129,145)(131,143)(133,141)(134,168)(135,139)(136,166)(138,164)(140,162)(142,160)(144,158)(146,156)(148,154)(150,152)(163,167)>;
G:=Group( (1,78,165)(2,79,166)(3,80,167)(4,81,168)(5,82,113)(6,83,114)(7,84,115)(8,85,116)(9,86,117)(10,87,118)(11,88,119)(12,89,120)(13,90,121)(14,91,122)(15,92,123)(16,93,124)(17,94,125)(18,95,126)(19,96,127)(20,97,128)(21,98,129)(22,99,130)(23,100,131)(24,101,132)(25,102,133)(26,103,134)(27,104,135)(28,105,136)(29,106,137)(30,107,138)(31,108,139)(32,109,140)(33,110,141)(34,111,142)(35,112,143)(36,57,144)(37,58,145)(38,59,146)(39,60,147)(40,61,148)(41,62,149)(42,63,150)(43,64,151)(44,65,152)(45,66,153)(46,67,154)(47,68,155)(48,69,156)(49,70,157)(50,71,158)(51,72,159)(52,73,160)(53,74,161)(54,75,162)(55,76,163)(56,77,164), (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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,71)(58,98)(59,69)(60,96)(61,67)(62,94)(63,65)(64,92)(66,90)(68,88)(70,86)(72,84)(73,111)(74,82)(75,109)(76,80)(77,107)(79,105)(81,103)(83,101)(85,99)(87,97)(89,95)(91,93)(100,112)(102,110)(104,108)(113,161)(114,132)(115,159)(116,130)(117,157)(118,128)(119,155)(120,126)(121,153)(122,124)(123,151)(125,149)(127,147)(129,145)(131,143)(133,141)(134,168)(135,139)(136,166)(138,164)(140,162)(142,160)(144,158)(146,156)(148,154)(150,152)(163,167) );
G=PermutationGroup([[(1,78,165),(2,79,166),(3,80,167),(4,81,168),(5,82,113),(6,83,114),(7,84,115),(8,85,116),(9,86,117),(10,87,118),(11,88,119),(12,89,120),(13,90,121),(14,91,122),(15,92,123),(16,93,124),(17,94,125),(18,95,126),(19,96,127),(20,97,128),(21,98,129),(22,99,130),(23,100,131),(24,101,132),(25,102,133),(26,103,134),(27,104,135),(28,105,136),(29,106,137),(30,107,138),(31,108,139),(32,109,140),(33,110,141),(34,111,142),(35,112,143),(36,57,144),(37,58,145),(38,59,146),(39,60,147),(40,61,148),(41,62,149),(42,63,150),(43,64,151),(44,65,152),(45,66,153),(46,67,154),(47,68,155),(48,69,156),(49,70,157),(50,71,158),(51,72,159),(52,73,160),(53,74,161),(54,75,162),(55,76,163),(56,77,164)], [(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)], [(2,28),(3,55),(4,26),(5,53),(6,24),(7,51),(8,22),(9,49),(10,20),(11,47),(12,18),(13,45),(14,16),(15,43),(17,41),(19,39),(21,37),(23,35),(25,33),(27,31),(30,56),(32,54),(34,52),(36,50),(38,48),(40,46),(42,44),(57,71),(58,98),(59,69),(60,96),(61,67),(62,94),(63,65),(64,92),(66,90),(68,88),(70,86),(72,84),(73,111),(74,82),(75,109),(76,80),(77,107),(79,105),(81,103),(83,101),(85,99),(87,97),(89,95),(91,93),(100,112),(102,110),(104,108),(113,161),(114,132),(115,159),(116,130),(117,157),(118,128),(119,155),(120,126),(121,153),(122,124),(123,151),(125,149),(127,147),(129,145),(131,143),(133,141),(134,168),(135,139),(136,166),(138,164),(140,162),(142,160),(144,158),(146,156),(148,154),(150,152),(163,167)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 28 | 1 | 1 | 2 | 28 | 1 | 1 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D7 | SD16 | C3×D4 | D14 | C3×D7 | C3×SD16 | D28 | C6×D7 | C56⋊C2 | C3×D28 | C3×C56⋊C2 |
| kernel | C3×C56⋊C2 | C168 | C3×Dic14 | C3×D28 | C56⋊C2 | C56 | Dic14 | D28 | C42 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C3×C56⋊C2 ►in GL2(𝔽337) generated by
| 208 | 0 |
| 0 | 208 |
| 123 | 177 |
| 160 | 159 |
| 1 | 0 |
| 143 | 336 |
G:=sub<GL(2,GF(337))| [208,0,0,208],[123,160,177,159],[1,143,0,336] >;
C3×C56⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_{56}\rtimes C_2 % in TeX
G:=Group("C3xC56:C2"); // GroupNames label
G:=SmallGroup(336,60);
// by ID
G=gap.SmallGroup(336,60);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,79,867,69,10373]);
// Polycyclic
G:=Group<a,b,c|a^3=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^27>;
// generators/relations
Export