metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊2D56, C21⋊2D8, D28⋊1S3, D84⋊5C2, C6.6D28, C42.2D4, C28.22D6, C12.2D14, C84.8C22, C3⋊C8⋊1D7, C7⋊1(D4⋊S3), C4.1(S3×D7), (C3×D28)⋊1C2, C14.1(C3⋊D4), C2.4(C3⋊D28), (C7×C3⋊C8)⋊1C2, SmallGroup(336,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D56
G = < a,b,c | a3=b56=c2=1, bab-1=cac=a-1, cbc=b-1 >
28C2
84C2
14C22
42C22
28C6
28S3
4D7
12D7
3C8
7D4
21D4
14C2×C6
14D6
2D14
6D14
4D21
21D8
7D12
3C56
3D28
2D42
3D56
Smallest permutation representation of C3⋊D56
►On 168 points
Generators in S168
(1 89 143)(2 144 90)(3 91 145)(4 146 92)(5 93 147)(6 148 94)(7 95 149)(8 150 96)(9 97 151)(10 152 98)(11 99 153)(12 154 100)(13 101 155)(14 156 102)(15 103 157)(16 158 104)(17 105 159)(18 160 106)(19 107 161)(20 162 108)(21 109 163)(22 164 110)(23 111 165)(24 166 112)(25 57 167)(26 168 58)(27 59 113)(28 114 60)(29 61 115)(30 116 62)(31 63 117)(32 118 64)(33 65 119)(34 120 66)(35 67 121)(36 122 68)(37 69 123)(38 124 70)(39 71 125)(40 126 72)(41 73 127)(42 128 74)(43 75 129)(44 130 76)(45 77 131)(46 132 78)(47 79 133)(48 134 80)(49 81 135)(50 136 82)(51 83 137)(52 138 84)(53 85 139)(54 140 86)(55 87 141)(56 142 88)
(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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 125)(58 124)(59 123)(60 122)(61 121)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 168)(71 167)(72 166)(73 165)(74 164)(75 163)(76 162)(77 161)(78 160)(79 159)(80 158)(81 157)(82 156)(83 155)(84 154)(85 153)(86 152)(87 151)(88 150)(89 149)(90 148)(91 147)(92 146)(93 145)(94 144)(95 143)(96 142)(97 141)(98 140)(99 139)(100 138)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)
G:=sub<Sym(168)| (1,89,143)(2,144,90)(3,91,145)(4,146,92)(5,93,147)(6,148,94)(7,95,149)(8,150,96)(9,97,151)(10,152,98)(11,99,153)(12,154,100)(13,101,155)(14,156,102)(15,103,157)(16,158,104)(17,105,159)(18,160,106)(19,107,161)(20,162,108)(21,109,163)(22,164,110)(23,111,165)(24,166,112)(25,57,167)(26,168,58)(27,59,113)(28,114,60)(29,61,115)(30,116,62)(31,63,117)(32,118,64)(33,65,119)(34,120,66)(35,67,121)(36,122,68)(37,69,123)(38,124,70)(39,71,125)(40,126,72)(41,73,127)(42,128,74)(43,75,129)(44,130,76)(45,77,131)(46,132,78)(47,79,133)(48,134,80)(49,81,135)(50,136,82)(51,83,137)(52,138,84)(53,85,139)(54,140,86)(55,87,141)(56,142,88), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)>;
G:=Group( (1,89,143)(2,144,90)(3,91,145)(4,146,92)(5,93,147)(6,148,94)(7,95,149)(8,150,96)(9,97,151)(10,152,98)(11,99,153)(12,154,100)(13,101,155)(14,156,102)(15,103,157)(16,158,104)(17,105,159)(18,160,106)(19,107,161)(20,162,108)(21,109,163)(22,164,110)(23,111,165)(24,166,112)(25,57,167)(26,168,58)(27,59,113)(28,114,60)(29,61,115)(30,116,62)(31,63,117)(32,118,64)(33,65,119)(34,120,66)(35,67,121)(36,122,68)(37,69,123)(38,124,70)(39,71,125)(40,126,72)(41,73,127)(42,128,74)(43,75,129)(44,130,76)(45,77,131)(46,132,78)(47,79,133)(48,134,80)(49,81,135)(50,136,82)(51,83,137)(52,138,84)(53,85,139)(54,140,86)(55,87,141)(56,142,88), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126) );
G=PermutationGroup([[(1,89,143),(2,144,90),(3,91,145),(4,146,92),(5,93,147),(6,148,94),(7,95,149),(8,150,96),(9,97,151),(10,152,98),(11,99,153),(12,154,100),(13,101,155),(14,156,102),(15,103,157),(16,158,104),(17,105,159),(18,160,106),(19,107,161),(20,162,108),(21,109,163),(22,164,110),(23,111,165),(24,166,112),(25,57,167),(26,168,58),(27,59,113),(28,114,60),(29,61,115),(30,116,62),(31,63,117),(32,118,64),(33,65,119),(34,120,66),(35,67,121),(36,122,68),(37,69,123),(38,124,70),(39,71,125),(40,126,72),(41,73,127),(42,128,74),(43,75,129),(44,130,76),(45,77,131),(46,132,78),(47,79,133),(48,134,80),(49,81,135),(50,136,82),(51,83,137),(52,138,84),(53,85,139),(54,140,86),(55,87,141),(56,142,88)], [(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)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(57,125),(58,124),(59,123),(60,122),(61,121),(62,120),(63,119),(64,118),(65,117),(66,116),(67,115),(68,114),(69,113),(70,168),(71,167),(72,166),(73,165),(74,164),(75,163),(76,162),(77,161),(78,160),(79,159),(80,158),(81,157),(82,156),(83,155),(84,154),(85,153),(86,152),(87,151),(88,150),(89,149),(90,148),(91,147),(92,146),(93,145),(94,144),(95,143),(96,142),(97,141),(98,140),(99,139),(100,138),(101,137),(102,136),(103,135),(104,134),(105,133),(106,132),(107,131),(108,130),(109,129),(110,128),(111,127),(112,126)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 12 | 14A | 14B | 14C | 21A | 21B | 21C | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 28 | 84 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 6 | 6 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | D8 | C3⋊D4 | D14 | D28 | D56 | D4⋊S3 | S3×D7 | C3⋊D28 | C3⋊D56 |
| kernel | C3⋊D56 | C7×C3⋊C8 | C3×D28 | D84 | D28 | C42 | C28 | C3⋊C8 | C21 | C14 | C12 | C6 | C3 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of C3⋊D56 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 112 |
| 0 | 0 | 9 | 335 |
| 237 | 174 | 0 | 0 |
| 163 | 50 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 328 | 1 |
| 275 | 195 | 0 | 0 |
| 281 | 62 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 336 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,1,9,0,0,112,335],[237,163,0,0,174,50,0,0,0,0,336,328,0,0,0,1],[275,281,0,0,195,62,0,0,0,0,1,9,0,0,0,336] >;
C3⋊D56 in GAP, Magma, Sage, TeX
C_3\rtimes D_{56} % in TeX
G:=Group("C3:D56"); // GroupNames label
G:=SmallGroup(336,30);
// by ID
G=gap.SmallGroup(336,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,79,218,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^3=b^56=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export