metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D21, C21⋊7D8, D84⋊2C2, C28.9D6, C4.1D42, C42.34D4, C12.9D14, C84.1C22, C7⋊3(D4⋊S3), C3⋊3(D4⋊D7), (C3×D4)⋊1D7, (C7×D4)⋊1S3, C21⋊C8⋊1C2, (D4×C21)⋊1C2, C6.16(C7⋊D4), C2.4(C21⋊7D4), C14.16(C3⋊D4), SmallGroup(336,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D21
G = < a,b,c,d | a4=b2=c21=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D21
►On 168 points
Generators in S168
(1 65 26 56)(2 66 27 57)(3 67 28 58)(4 68 29 59)(5 69 30 60)(6 70 31 61)(7 71 32 62)(8 72 33 63)(9 73 34 43)(10 74 35 44)(11 75 36 45)(12 76 37 46)(13 77 38 47)(14 78 39 48)(15 79 40 49)(16 80 41 50)(17 81 42 51)(18 82 22 52)(19 83 23 53)(20 84 24 54)(21 64 25 55)(85 142 116 159)(86 143 117 160)(87 144 118 161)(88 145 119 162)(89 146 120 163)(90 147 121 164)(91 127 122 165)(92 128 123 166)(93 129 124 167)(94 130 125 168)(95 131 126 148)(96 132 106 149)(97 133 107 150)(98 134 108 151)(99 135 109 152)(100 136 110 153)(101 137 111 154)(102 138 112 155)(103 139 113 156)(104 140 114 157)(105 141 115 158)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)(57 104)(58 105)(59 85)(60 86)(61 87)(62 88)(63 89)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 29)(23 28)(24 27)(25 26)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(85 135)(86 134)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 147)(95 146)(96 145)(97 144)(98 143)(99 142)(100 141)(101 140)(102 139)(103 138)(104 137)(105 136)(106 162)(107 161)(108 160)(109 159)(110 158)(111 157)(112 156)(113 155)(114 154)(115 153)(116 152)(117 151)(118 150)(119 149)(120 148)(121 168)(122 167)(123 166)(124 165)(125 164)(126 163)
G:=sub<Sym(168)| (1,65,26,56)(2,66,27,57)(3,67,28,58)(4,68,29,59)(5,69,30,60)(6,70,31,61)(7,71,32,62)(8,72,33,63)(9,73,34,43)(10,74,35,44)(11,75,36,45)(12,76,37,46)(13,77,38,47)(14,78,39,48)(15,79,40,49)(16,80,41,50)(17,81,42,51)(18,82,22,52)(19,83,23,53)(20,84,24,54)(21,64,25,55)(85,142,116,159)(86,143,117,160)(87,144,118,161)(88,145,119,162)(89,146,120,163)(90,147,121,164)(91,127,122,165)(92,128,123,166)(93,129,124,167)(94,130,125,168)(95,131,126,148)(96,132,106,149)(97,133,107,150)(98,134,108,151)(99,135,109,152)(100,136,110,153)(101,137,111,154)(102,138,112,155)(103,139,113,156)(104,140,114,157)(105,141,115,158), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,85)(60,86)(61,87)(62,88)(63,89)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,29)(23,28)(24,27)(25,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163)>;
G:=Group( (1,65,26,56)(2,66,27,57)(3,67,28,58)(4,68,29,59)(5,69,30,60)(6,70,31,61)(7,71,32,62)(8,72,33,63)(9,73,34,43)(10,74,35,44)(11,75,36,45)(12,76,37,46)(13,77,38,47)(14,78,39,48)(15,79,40,49)(16,80,41,50)(17,81,42,51)(18,82,22,52)(19,83,23,53)(20,84,24,54)(21,64,25,55)(85,142,116,159)(86,143,117,160)(87,144,118,161)(88,145,119,162)(89,146,120,163)(90,147,121,164)(91,127,122,165)(92,128,123,166)(93,129,124,167)(94,130,125,168)(95,131,126,148)(96,132,106,149)(97,133,107,150)(98,134,108,151)(99,135,109,152)(100,136,110,153)(101,137,111,154)(102,138,112,155)(103,139,113,156)(104,140,114,157)(105,141,115,158), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103)(57,104)(58,105)(59,85)(60,86)(61,87)(62,88)(63,89)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,29)(23,28)(24,27)(25,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(85,135)(86,134)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163) );
G=PermutationGroup([[(1,65,26,56),(2,66,27,57),(3,67,28,58),(4,68,29,59),(5,69,30,60),(6,70,31,61),(7,71,32,62),(8,72,33,63),(9,73,34,43),(10,74,35,44),(11,75,36,45),(12,76,37,46),(13,77,38,47),(14,78,39,48),(15,79,40,49),(16,80,41,50),(17,81,42,51),(18,82,22,52),(19,83,23,53),(20,84,24,54),(21,64,25,55),(85,142,116,159),(86,143,117,160),(87,144,118,161),(88,145,119,162),(89,146,120,163),(90,147,121,164),(91,127,122,165),(92,128,123,166),(93,129,124,167),(94,130,125,168),(95,131,126,148),(96,132,106,149),(97,133,107,150),(98,134,108,151),(99,135,109,152),(100,136,110,153),(101,137,111,154),(102,138,112,155),(103,139,113,156),(104,140,114,157),(105,141,115,158)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,161),(7,162),(8,163),(9,164),(10,165),(11,166),(12,167),(13,168),(14,148),(15,149),(16,150),(17,151),(18,152),(19,153),(20,154),(21,155),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,133),(42,134),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103),(57,104),(58,105),(59,85),(60,86),(61,87),(62,88),(63,89),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,29),(23,28),(24,27),(25,26),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(85,135),(86,134),(87,133),(88,132),(89,131),(90,130),(91,129),(92,128),(93,127),(94,147),(95,146),(96,145),(97,144),(98,143),(99,142),(100,141),(101,140),(102,139),(103,138),(104,137),(105,136),(106,162),(107,161),(108,160),(109,159),(110,158),(111,157),(112,156),(113,155),(114,154),(115,153),(116,152),(117,151),(118,150),(119,149),(120,148),(121,168),(122,167),(123,166),(124,165),(125,164),(126,163)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 12 | 14A | 14B | 14C | 14D | ··· | 14I | 21A | ··· | 21F | 28A | 28B | 28C | 42A | ··· | 42F | 42G | ··· | 42R | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | 28 | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 4 | 84 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 42 | 42 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | D8 | C3⋊D4 | D14 | D21 | C7⋊D4 | D42 | C21⋊7D4 | D4⋊S3 | D4⋊D7 | D4⋊D21 |
| kernel | D4⋊D21 | C21⋊C8 | D84 | D4×C21 | C7×D4 | C42 | C28 | C3×D4 | C21 | C14 | C12 | D4 | C6 | C4 | C2 | C7 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 1 | 3 | 6 |
Matrix representation of D4⋊D21 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 47 |
| 0 | 0 | 86 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 63 |
| 0 | 0 | 230 | 311 |
| 180 | 84 | 0 | 0 |
| 169 | 77 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 142 | 109 | 0 | 0 |
| 84 | 195 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 251 | 336 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,86,0,0,47,1],[1,0,0,0,0,1,0,0,0,0,26,230,0,0,63,311],[180,169,0,0,84,77,0,0,0,0,1,0,0,0,0,1],[142,84,0,0,109,195,0,0,0,0,1,251,0,0,0,336] >;
D4⋊D21 in GAP, Magma, Sage, TeX
D_4\rtimes D_{21} % in TeX
G:=Group("D4:D21"); // GroupNames label
G:=SmallGroup(336,101);
// by ID
G=gap.SmallGroup(336,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,218,116,50,964,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^21=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export