metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D84⋊6C2, D6.6D14, C28.28D6, Dic14⋊5S3, C12.17D14, C42.6C23, Dic7.3D6, C84.14C22, D42.1C22, Dic3.11D14, (C4×S3)⋊2D7, C4.7(S3×D7), (S3×C28)⋊2C2, D21⋊C4⋊1C2, C21⋊4(C4○D4), C3⋊1(C4○D28), C7⋊D12⋊2C2, C7⋊1(Q8⋊3S3), C6.6(C22×D7), (C3×Dic14)⋊3C2, C14.6(C22×S3), (S3×C14).7C22, (C3×Dic7).3C22, (C7×Dic3).9C22, C2.10(C2×S3×D7), SmallGroup(336,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D84⋊C2
G = < a,b,c | a84=b2=c2=1, bab=a-1, cac=a29, cbc=a70b >
Subgroups: 484 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, D7, C14, C14, C4○D4, C21, C4×S3, C4×S3, D12, C3×Q8, Dic7, C28, C28, D14, C2×C14, S3×C7, D21, C42, Q8⋊3S3, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×Dic3, C3×Dic7, C84, S3×C14, D42, C4○D28, D21⋊C4, C7⋊D12, C3×Dic14, S3×C28, D84, D84⋊C2
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, Q8⋊3S3, C22×D7, S3×D7, C4○D28, C2×S3×D7, D84⋊C2
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(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)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(102 168)(103 167)(104 166)(105 165)(106 164)(107 163)(108 162)(109 161)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)
(1 167)(2 112)(3 141)(4 86)(5 115)(6 144)(7 89)(8 118)(9 147)(10 92)(11 121)(12 150)(13 95)(14 124)(15 153)(16 98)(17 127)(18 156)(19 101)(20 130)(21 159)(22 104)(23 133)(24 162)(25 107)(26 136)(27 165)(28 110)(29 139)(30 168)(31 113)(32 142)(33 87)(34 116)(35 145)(36 90)(37 119)(38 148)(39 93)(40 122)(41 151)(42 96)(43 125)(44 154)(45 99)(46 128)(47 157)(48 102)(49 131)(50 160)(51 105)(52 134)(53 163)(54 108)(55 137)(56 166)(57 111)(58 140)(59 85)(60 114)(61 143)(62 88)(63 117)(64 146)(65 91)(66 120)(67 149)(68 94)(69 123)(70 152)(71 97)(72 126)(73 155)(74 100)(75 129)(76 158)(77 103)(78 132)(79 161)(80 106)(81 135)(82 164)(83 109)(84 138)
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(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)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(102,168)(103,167)(104,166)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136), (1,167)(2,112)(3,141)(4,86)(5,115)(6,144)(7,89)(8,118)(9,147)(10,92)(11,121)(12,150)(13,95)(14,124)(15,153)(16,98)(17,127)(18,156)(19,101)(20,130)(21,159)(22,104)(23,133)(24,162)(25,107)(26,136)(27,165)(28,110)(29,139)(30,168)(31,113)(32,142)(33,87)(34,116)(35,145)(36,90)(37,119)(38,148)(39,93)(40,122)(41,151)(42,96)(43,125)(44,154)(45,99)(46,128)(47,157)(48,102)(49,131)(50,160)(51,105)(52,134)(53,163)(54,108)(55,137)(56,166)(57,111)(58,140)(59,85)(60,114)(61,143)(62,88)(63,117)(64,146)(65,91)(66,120)(67,149)(68,94)(69,123)(70,152)(71,97)(72,126)(73,155)(74,100)(75,129)(76,158)(77,103)(78,132)(79,161)(80,106)(81,135)(82,164)(83,109)(84,138)>;
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(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)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(102,168)(103,167)(104,166)(105,165)(106,164)(107,163)(108,162)(109,161)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136), (1,167)(2,112)(3,141)(4,86)(5,115)(6,144)(7,89)(8,118)(9,147)(10,92)(11,121)(12,150)(13,95)(14,124)(15,153)(16,98)(17,127)(18,156)(19,101)(20,130)(21,159)(22,104)(23,133)(24,162)(25,107)(26,136)(27,165)(28,110)(29,139)(30,168)(31,113)(32,142)(33,87)(34,116)(35,145)(36,90)(37,119)(38,148)(39,93)(40,122)(41,151)(42,96)(43,125)(44,154)(45,99)(46,128)(47,157)(48,102)(49,131)(50,160)(51,105)(52,134)(53,163)(54,108)(55,137)(56,166)(57,111)(58,140)(59,85)(60,114)(61,143)(62,88)(63,117)(64,146)(65,91)(66,120)(67,149)(68,94)(69,123)(70,152)(71,97)(72,126)(73,155)(74,100)(75,129)(76,158)(77,103)(78,132)(79,161)(80,106)(81,135)(82,164)(83,109)(84,138) );
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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(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),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(102,168),(103,167),(104,166),(105,165),(106,164),(107,163),(108,162),(109,161),(110,160),(111,159),(112,158),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,143),(128,142),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136)], [(1,167),(2,112),(3,141),(4,86),(5,115),(6,144),(7,89),(8,118),(9,147),(10,92),(11,121),(12,150),(13,95),(14,124),(15,153),(16,98),(17,127),(18,156),(19,101),(20,130),(21,159),(22,104),(23,133),(24,162),(25,107),(26,136),(27,165),(28,110),(29,139),(30,168),(31,113),(32,142),(33,87),(34,116),(35,145),(36,90),(37,119),(38,148),(39,93),(40,122),(41,151),(42,96),(43,125),(44,154),(45,99),(46,128),(47,157),(48,102),(49,131),(50,160),(51,105),(52,134),(53,163),(54,108),(55,137),(56,166),(57,111),(58,140),(59,85),(60,114),(61,143),(62,88),(63,117),(64,146),(65,91),(66,120),(67,149),(68,94),(69,123),(70,152),(71,97),(72,126),(73,155),(74,100),(75,129),(76,158),(77,103),(78,132),(79,161),(80,106),(81,135),(82,164),(83,109),(84,138)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 7A | 7B | 7C | 12A | 12B | 12C | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | ··· | 28F | 28G | ··· | 28L | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 42 | 42 | 2 | 2 | 3 | 3 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | Q8⋊3S3 | S3×D7 | C2×S3×D7 | D84⋊C2 |
| kernel | D84⋊C2 | D21⋊C4 | C7⋊D12 | C3×Dic14 | S3×C28 | D84 | Dic14 | Dic7 | C28 | C4×S3 | C21 | Dic3 | C12 | D6 | C3 | C7 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of D84⋊C2 ►in GL6(𝔽337)
| 336 | 335 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 193 | 0 | 0 |
| 0 | 0 | 252 | 228 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 |
| 0 | 0 | 0 | 0 | 1 | 336 |
| 336 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 228 | 336 | 0 | 0 |
| 0 | 0 | 85 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 336 |
| 0 | 0 | 0 | 0 | 0 | 336 |
| 189 | 41 | 0 | 0 | 0 | 0 |
| 148 | 148 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(337))| [336,1,0,0,0,0,335,1,0,0,0,0,0,0,336,252,0,0,0,0,193,228,0,0,0,0,0,0,0,1,0,0,0,0,336,336],[336,1,0,0,0,0,0,1,0,0,0,0,0,0,228,85,0,0,0,0,336,109,0,0,0,0,0,0,1,0,0,0,0,0,336,336],[189,148,0,0,0,0,41,148,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D84⋊C2 in GAP, Magma, Sage, TeX
D_{84}\rtimes C_2 % in TeX
G:=Group("D84:C2"); // GroupNames label
G:=SmallGroup(336,142);
// by ID
G=gap.SmallGroup(336,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,55,218,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^84=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^29,c*b*c=a^70*b>;
// generators/relations