direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C21⋊D4, C42⋊1D4, D14⋊5D6, D6⋊5D14, C42.21C23, Dic21⋊9C22, C21⋊4(C2×D4), C6⋊2(C7⋊D4), C14⋊2(C3⋊D4), (C2×C14).16D6, (C2×C6).16D14, (C22×S3)⋊1D7, (C6×D7)⋊5C22, (C22×D7)⋊2S3, (S3×C14)⋊5C22, C22.14(S3×D7), C6.21(C22×D7), (C2×Dic21)⋊10C2, (C2×C42).15C22, C14.21(C22×S3), (C2×C6×D7)⋊1C2, C7⋊3(C2×C3⋊D4), C3⋊3(C2×C7⋊D4), (S3×C2×C14)⋊1C2, C2.21(C2×S3×D7), SmallGroup(336,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C21⋊D4
G = < a,b,c,d | a2=b21=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b13, dcd=c-1 >
Subgroups: 540 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, Dic3, D6, D6, C2×C6, C2×C6, D7, C14, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, Dic7, D14, D14, C2×C14, C2×C14, S3×C7, C3×D7, C42, C42, C2×C3⋊D4, C2×Dic7, C7⋊D4, C22×D7, C22×C14, Dic21, C6×D7, C6×D7, S3×C14, S3×C14, C2×C42, C2×C7⋊D4, C21⋊D4, C2×Dic21, C2×C6×D7, S3×C2×C14, C2×C21⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, C2×C3⋊D4, C7⋊D4, C22×D7, S3×D7, C2×C7⋊D4, C21⋊D4, C2×S3×D7, C2×C21⋊D4
►On 168 points
Generators in S168
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 150)(107 151)(108 152)(109 153)(110 154)(111 155)(112 156)(113 157)(114 158)(115 159)(116 160)(117 161)(118 162)(119 163)(120 164)(121 165)(122 166)(123 167)(124 168)(125 148)(126 149)
(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 98 33 117)(2 97 34 116)(3 96 35 115)(4 95 36 114)(5 94 37 113)(6 93 38 112)(7 92 39 111)(8 91 40 110)(9 90 41 109)(10 89 42 108)(11 88 22 107)(12 87 23 106)(13 86 24 126)(14 85 25 125)(15 105 26 124)(16 104 27 123)(17 103 28 122)(18 102 29 121)(19 101 30 120)(20 100 31 119)(21 99 32 118)(43 130 81 161)(44 129 82 160)(45 128 83 159)(46 127 84 158)(47 147 64 157)(48 146 65 156)(49 145 66 155)(50 144 67 154)(51 143 68 153)(52 142 69 152)(53 141 70 151)(54 140 71 150)(55 139 72 149)(56 138 73 148)(57 137 74 168)(58 136 75 167)(59 135 76 166)(60 134 77 165)(61 133 78 164)(62 132 79 163)(63 131 80 162)
(1 43)(2 56)(3 48)(4 61)(5 53)(6 45)(7 58)(8 50)(9 63)(10 55)(11 47)(12 60)(13 52)(14 44)(15 57)(16 49)(17 62)(18 54)(19 46)(20 59)(21 51)(22 64)(23 77)(24 69)(25 82)(26 74)(27 66)(28 79)(29 71)(30 84)(31 76)(32 68)(33 81)(34 73)(35 65)(36 78)(37 70)(38 83)(39 75)(40 67)(41 80)(42 72)(85 160)(86 152)(87 165)(88 157)(89 149)(90 162)(91 154)(92 167)(93 159)(94 151)(95 164)(96 156)(97 148)(98 161)(99 153)(100 166)(101 158)(102 150)(103 163)(104 155)(105 168)(106 134)(107 147)(108 139)(109 131)(110 144)(111 136)(112 128)(113 141)(114 133)(115 146)(116 138)(117 130)(118 143)(119 135)(120 127)(121 140)(122 132)(123 145)(124 137)(125 129)(126 142)
G:=sub<Sym(168)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,148)(126,149), (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), (1,98,33,117)(2,97,34,116)(3,96,35,115)(4,95,36,114)(5,94,37,113)(6,93,38,112)(7,92,39,111)(8,91,40,110)(9,90,41,109)(10,89,42,108)(11,88,22,107)(12,87,23,106)(13,86,24,126)(14,85,25,125)(15,105,26,124)(16,104,27,123)(17,103,28,122)(18,102,29,121)(19,101,30,120)(20,100,31,119)(21,99,32,118)(43,130,81,161)(44,129,82,160)(45,128,83,159)(46,127,84,158)(47,147,64,157)(48,146,65,156)(49,145,66,155)(50,144,67,154)(51,143,68,153)(52,142,69,152)(53,141,70,151)(54,140,71,150)(55,139,72,149)(56,138,73,148)(57,137,74,168)(58,136,75,167)(59,135,76,166)(60,134,77,165)(61,133,78,164)(62,132,79,163)(63,131,80,162), (1,43)(2,56)(3,48)(4,61)(5,53)(6,45)(7,58)(8,50)(9,63)(10,55)(11,47)(12,60)(13,52)(14,44)(15,57)(16,49)(17,62)(18,54)(19,46)(20,59)(21,51)(22,64)(23,77)(24,69)(25,82)(26,74)(27,66)(28,79)(29,71)(30,84)(31,76)(32,68)(33,81)(34,73)(35,65)(36,78)(37,70)(38,83)(39,75)(40,67)(41,80)(42,72)(85,160)(86,152)(87,165)(88,157)(89,149)(90,162)(91,154)(92,167)(93,159)(94,151)(95,164)(96,156)(97,148)(98,161)(99,153)(100,166)(101,158)(102,150)(103,163)(104,155)(105,168)(106,134)(107,147)(108,139)(109,131)(110,144)(111,136)(112,128)(113,141)(114,133)(115,146)(116,138)(117,130)(118,143)(119,135)(120,127)(121,140)(122,132)(123,145)(124,137)(125,129)(126,142)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,148)(126,149), (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), (1,98,33,117)(2,97,34,116)(3,96,35,115)(4,95,36,114)(5,94,37,113)(6,93,38,112)(7,92,39,111)(8,91,40,110)(9,90,41,109)(10,89,42,108)(11,88,22,107)(12,87,23,106)(13,86,24,126)(14,85,25,125)(15,105,26,124)(16,104,27,123)(17,103,28,122)(18,102,29,121)(19,101,30,120)(20,100,31,119)(21,99,32,118)(43,130,81,161)(44,129,82,160)(45,128,83,159)(46,127,84,158)(47,147,64,157)(48,146,65,156)(49,145,66,155)(50,144,67,154)(51,143,68,153)(52,142,69,152)(53,141,70,151)(54,140,71,150)(55,139,72,149)(56,138,73,148)(57,137,74,168)(58,136,75,167)(59,135,76,166)(60,134,77,165)(61,133,78,164)(62,132,79,163)(63,131,80,162), (1,43)(2,56)(3,48)(4,61)(5,53)(6,45)(7,58)(8,50)(9,63)(10,55)(11,47)(12,60)(13,52)(14,44)(15,57)(16,49)(17,62)(18,54)(19,46)(20,59)(21,51)(22,64)(23,77)(24,69)(25,82)(26,74)(27,66)(28,79)(29,71)(30,84)(31,76)(32,68)(33,81)(34,73)(35,65)(36,78)(37,70)(38,83)(39,75)(40,67)(41,80)(42,72)(85,160)(86,152)(87,165)(88,157)(89,149)(90,162)(91,154)(92,167)(93,159)(94,151)(95,164)(96,156)(97,148)(98,161)(99,153)(100,166)(101,158)(102,150)(103,163)(104,155)(105,168)(106,134)(107,147)(108,139)(109,131)(110,144)(111,136)(112,128)(113,141)(114,133)(115,146)(116,138)(117,130)(118,143)(119,135)(120,127)(121,140)(122,132)(123,145)(124,137)(125,129)(126,142) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132),(101,133),(102,134),(103,135),(104,136),(105,137),(106,150),(107,151),(108,152),(109,153),(110,154),(111,155),(112,156),(113,157),(114,158),(115,159),(116,160),(117,161),(118,162),(119,163),(120,164),(121,165),(122,166),(123,167),(124,168),(125,148),(126,149)], [(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)], [(1,98,33,117),(2,97,34,116),(3,96,35,115),(4,95,36,114),(5,94,37,113),(6,93,38,112),(7,92,39,111),(8,91,40,110),(9,90,41,109),(10,89,42,108),(11,88,22,107),(12,87,23,106),(13,86,24,126),(14,85,25,125),(15,105,26,124),(16,104,27,123),(17,103,28,122),(18,102,29,121),(19,101,30,120),(20,100,31,119),(21,99,32,118),(43,130,81,161),(44,129,82,160),(45,128,83,159),(46,127,84,158),(47,147,64,157),(48,146,65,156),(49,145,66,155),(50,144,67,154),(51,143,68,153),(52,142,69,152),(53,141,70,151),(54,140,71,150),(55,139,72,149),(56,138,73,148),(57,137,74,168),(58,136,75,167),(59,135,76,166),(60,134,77,165),(61,133,78,164),(62,132,79,163),(63,131,80,162)], [(1,43),(2,56),(3,48),(4,61),(5,53),(6,45),(7,58),(8,50),(9,63),(10,55),(11,47),(12,60),(13,52),(14,44),(15,57),(16,49),(17,62),(18,54),(19,46),(20,59),(21,51),(22,64),(23,77),(24,69),(25,82),(26,74),(27,66),(28,79),(29,71),(30,84),(31,76),(32,68),(33,81),(34,73),(35,65),(36,78),(37,70),(38,83),(39,75),(40,67),(41,80),(42,72),(85,160),(86,152),(87,165),(88,157),(89,149),(90,162),(91,154),(92,167),(93,159),(94,151),(95,164),(96,156),(97,148),(98,161),(99,153),(100,166),(101,158),(102,150),(103,163),(104,155),(105,168),(106,134),(107,147),(108,139),(109,131),(110,144),(111,136),(112,128),(113,141),(114,133),(115,146),(116,138),(117,130),(118,143),(119,135),(120,127),(121,140),(122,132),(123,145),(124,137),(125,129),(126,142)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 21A | 21B | 21C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 14 | 14 | 2 | 42 | 42 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D7 | C3⋊D4 | D14 | D14 | C7⋊D4 | S3×D7 | C21⋊D4 | C2×S3×D7 |
| kernel | C2×C21⋊D4 | C21⋊D4 | C2×Dic21 | C2×C6×D7 | S3×C2×C14 | C22×D7 | C42 | D14 | C2×C14 | C22×S3 | C14 | D6 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 | 3 | 6 | 3 |
Matrix representation of C2×C21⋊D4 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 336 | 143 | 0 | 0 |
| 194 | 228 | 0 | 0 |
| 0 | 0 | 128 | 0 |
| 0 | 0 | 101 | 208 |
| 305 | 90 | 0 | 0 |
| 232 | 32 | 0 | 0 |
| 0 | 0 | 230 | 2 |
| 0 | 0 | 4 | 107 |
| 336 | 0 | 0 | 0 |
| 194 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 107 | 336 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1],[336,194,0,0,143,228,0,0,0,0,128,101,0,0,0,208],[305,232,0,0,90,32,0,0,0,0,230,4,0,0,2,107],[336,194,0,0,0,1,0,0,0,0,1,107,0,0,0,336] >;
C2×C21⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{21}\rtimes D_4 % in TeX
G:=Group("C2xC21:D4"); // GroupNames label
G:=SmallGroup(336,157);
// by ID
G=gap.SmallGroup(336,157);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^21=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^13,d*c*d=c^-1>;
// generators/relations