direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D21, C28⋊8D6, C12⋊8D14, C84⋊9C22, C22.9D42, C42.29C23, D42.15C22, Dic21⋊10C22, C6⋊2(C4×D7), C14⋊2(C4×S3), (C2×C84)⋊7C2, (C2×C28)⋊5S3, C42⋊4(C2×C4), (C2×C12)⋊5D7, C21⋊5(C22×C4), (C2×C6).27D14, (C2×C14).27D6, C2.1(C22×D21), C6.29(C22×D7), (C2×Dic21)⋊11C2, (C2×C42).28C22, C14.29(C22×S3), (C22×D21).4C2, C7⋊3(S3×C2×C4), C3⋊3(C2×C4×D7), SmallGroup(336,195)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C2×C4×D21 |
Generators and relations for C2×C4×D21
G = < a,b,c,d | a2=b4=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 640 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, D7, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic7, C28, D14, C2×C14, D21, C42, C42, S3×C2×C4, C4×D7, C2×Dic7, C2×C28, C22×D7, Dic21, C84, D42, C2×C42, C2×C4×D7, C4×D21, C2×Dic21, C2×C84, C22×D21, C2×C4×D21
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, D7, C22×C4, C4×S3, C22×S3, D14, D21, S3×C2×C4, C4×D7, C22×D7, D42, C2×C4×D7, C4×D21, C22×D21, C2×C4×D21
►On 168 points
Generators in S168
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)
(1 65 32 61)(2 66 33 62)(3 67 34 63)(4 68 35 43)(5 69 36 44)(6 70 37 45)(7 71 38 46)(8 72 39 47)(9 73 40 48)(10 74 41 49)(11 75 42 50)(12 76 22 51)(13 77 23 52)(14 78 24 53)(15 79 25 54)(16 80 26 55)(17 81 27 56)(18 82 28 57)(19 83 29 58)(20 84 30 59)(21 64 31 60)(85 156 117 129)(86 157 118 130)(87 158 119 131)(88 159 120 132)(89 160 121 133)(90 161 122 134)(91 162 123 135)(92 163 124 136)(93 164 125 137)(94 165 126 138)(95 166 106 139)(96 167 107 140)(97 168 108 141)(98 148 109 142)(99 149 110 143)(100 150 111 144)(101 151 112 145)(102 152 113 146)(103 153 114 147)(104 154 115 127)(105 155 116 128)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(43 82)(44 81)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 84)(63 83)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(127 152)(128 151)(129 150)(130 149)(131 148)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)
G:=sub<Sym(168)| (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162), (1,65,32,61)(2,66,33,62)(3,67,34,63)(4,68,35,43)(5,69,36,44)(6,70,37,45)(7,71,38,46)(8,72,39,47)(9,73,40,48)(10,74,41,49)(11,75,42,50)(12,76,22,51)(13,77,23,52)(14,78,24,53)(15,79,25,54)(16,80,26,55)(17,81,27,56)(18,82,28,57)(19,83,29,58)(20,84,30,59)(21,64,31,60)(85,156,117,129)(86,157,118,130)(87,158,119,131)(88,159,120,132)(89,160,121,133)(90,161,122,134)(91,162,123,135)(92,163,124,136)(93,164,125,137)(94,165,126,138)(95,166,106,139)(96,167,107,140)(97,168,108,141)(98,148,109,142)(99,149,110,143)(100,150,111,144)(101,151,112,145)(102,152,113,146)(103,153,114,147)(104,154,115,127)(105,155,116,128), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,84)(63,83)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(127,152)(128,151)(129,150)(130,149)(131,148)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)>;
G:=Group( (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162), (1,65,32,61)(2,66,33,62)(3,67,34,63)(4,68,35,43)(5,69,36,44)(6,70,37,45)(7,71,38,46)(8,72,39,47)(9,73,40,48)(10,74,41,49)(11,75,42,50)(12,76,22,51)(13,77,23,52)(14,78,24,53)(15,79,25,54)(16,80,26,55)(17,81,27,56)(18,82,28,57)(19,83,29,58)(20,84,30,59)(21,64,31,60)(85,156,117,129)(86,157,118,130)(87,158,119,131)(88,159,120,132)(89,160,121,133)(90,161,122,134)(91,162,123,135)(92,163,124,136)(93,164,125,137)(94,165,126,138)(95,166,106,139)(96,167,107,140)(97,168,108,141)(98,148,109,142)(99,149,110,143)(100,150,111,144)(101,151,112,145)(102,152,113,146)(103,153,114,147)(104,154,115,127)(105,155,116,128), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,84)(63,83)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(127,152)(128,151)(129,150)(130,149)(131,148)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153) );
G=PermutationGroup([[(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,105),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,121),(29,122),(30,123),(31,124),(32,125),(33,126),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,140),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,127),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,163),(65,164),(66,165),(67,166),(68,167),(69,168),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162)], [(1,65,32,61),(2,66,33,62),(3,67,34,63),(4,68,35,43),(5,69,36,44),(6,70,37,45),(7,71,38,46),(8,72,39,47),(9,73,40,48),(10,74,41,49),(11,75,42,50),(12,76,22,51),(13,77,23,52),(14,78,24,53),(15,79,25,54),(16,80,26,55),(17,81,27,56),(18,82,28,57),(19,83,29,58),(20,84,30,59),(21,64,31,60),(85,156,117,129),(86,157,118,130),(87,158,119,131),(88,159,120,132),(89,160,121,133),(90,161,122,134),(91,162,123,135),(92,163,124,136),(93,164,125,137),(94,165,126,138),(95,166,106,139),(96,167,107,140),(97,168,108,141),(98,148,109,142),(99,149,110,143),(100,150,111,144),(101,151,112,145),(102,152,113,146),(103,153,114,147),(104,154,115,127),(105,155,116,128)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(43,82),(44,81),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,84),(63,83),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112),(127,152),(128,151),(129,150),(130,149),(131,148),(132,168),(133,167),(134,166),(135,165),(136,164),(137,163),(138,162),(139,161),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 21 | 21 | 21 | 21 | 2 | 1 | 1 | 1 | 1 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D7 | C4×S3 | D14 | D14 | D21 | C4×D7 | D42 | D42 | C4×D21 |
| kernel | C2×C4×D21 | C4×D21 | C2×Dic21 | C2×C84 | C22×D21 | D42 | C2×C28 | C28 | C2×C14 | C2×C12 | C14 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of C2×C4×D21 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 189 | 0 | 0 | 0 |
| 0 | 189 | 0 | 0 |
| 0 | 0 | 148 | 0 |
| 0 | 0 | 0 | 148 |
| 76 | 110 | 0 | 0 |
| 117 | 227 | 0 | 0 |
| 0 | 0 | 70 | 323 |
| 0 | 0 | 14 | 262 |
| 226 | 33 | 0 | 0 |
| 76 | 111 | 0 | 0 |
| 0 | 0 | 304 | 110 |
| 0 | 0 | 33 | 33 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[189,0,0,0,0,189,0,0,0,0,148,0,0,0,0,148],[76,117,0,0,110,227,0,0,0,0,70,14,0,0,323,262],[226,76,0,0,33,111,0,0,0,0,304,33,0,0,110,33] >;
C2×C4×D21 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{21} % in TeX
G:=Group("C2xC4xD21"); // GroupNames label
G:=SmallGroup(336,195);
// by ID
G=gap.SmallGroup(336,195);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,50,964,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^21=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations