direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C21⋊7D4, C42⋊7D4, C23⋊2D21, C22⋊3D42, D42⋊7C22, C42.37C23, Dic21⋊4C22, (C2×C6)⋊8D14, C21⋊16(C2×D4), (C2×C14)⋊11D6, C6⋊3(C7⋊D4), (C22×C6)⋊2D7, C14⋊3(C3⋊D4), (C22×C14)⋊4S3, (C22×C42)⋊2C2, (C2×C42)⋊9C22, (C2×Dic21)⋊4C2, (C22×D21)⋊3C2, C6.37(C22×D7), C14.37(C22×S3), C2.10(C22×D21), C7⋊4(C2×C3⋊D4), C3⋊4(C2×C7⋊D4), SmallGroup(336,203)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C21⋊7D4
G = < a,b,c,d | a2=b21=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 672 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C7, C2×C4, D4, C23, C23, Dic3, D6, C2×C6, C2×C6, C2×C6, D7, C14, C14, C14, C2×D4, C21, C2×Dic3, C3⋊D4, C22×S3, C22×C6, Dic7, D14, C2×C14, C2×C14, C2×C14, D21, C42, C42, C42, C2×C3⋊D4, C2×Dic7, C7⋊D4, C22×D7, C22×C14, Dic21, D42, D42, C2×C42, C2×C42, C2×C42, C2×C7⋊D4, C2×Dic21, C21⋊7D4, C22×D21, C22×C42, C2×C21⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C3⋊D4, C22×S3, D14, D21, C2×C3⋊D4, C7⋊D4, C22×D7, D42, C2×C7⋊D4, C21⋊7D4, C22×D21, C2×C21⋊7D4
►On 168 points
Generators in S168
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 144)(44 145)(45 146)(46 147)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)
(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 56 35 75)(2 55 36 74)(3 54 37 73)(4 53 38 72)(5 52 39 71)(6 51 40 70)(7 50 41 69)(8 49 42 68)(9 48 22 67)(10 47 23 66)(11 46 24 65)(12 45 25 64)(13 44 26 84)(14 43 27 83)(15 63 28 82)(16 62 29 81)(17 61 30 80)(18 60 31 79)(19 59 32 78)(20 58 33 77)(21 57 34 76)(85 145 118 153)(86 144 119 152)(87 143 120 151)(88 142 121 150)(89 141 122 149)(90 140 123 148)(91 139 124 168)(92 138 125 167)(93 137 126 166)(94 136 106 165)(95 135 107 164)(96 134 108 163)(97 133 109 162)(98 132 110 161)(99 131 111 160)(100 130 112 159)(101 129 113 158)(102 128 114 157)(103 127 115 156)(104 147 116 155)(105 146 117 154)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 105)(12 104)(13 103)(14 102)(15 101)(16 100)(17 99)(18 98)(19 97)(20 96)(21 95)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 168)(54 167)(55 166)(56 165)(57 164)(58 163)(59 162)(60 161)(61 160)(62 159)(63 158)(64 147)(65 146)(66 145)(67 144)(68 143)(69 142)(70 141)(71 140)(72 139)(73 138)(74 137)(75 136)(76 135)(77 134)(78 133)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)
G:=sub<Sym(168)| (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,144)(44,145)(45,146)(46,147)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153), (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), (1,56,35,75)(2,55,36,74)(3,54,37,73)(4,53,38,72)(5,52,39,71)(6,51,40,70)(7,50,41,69)(8,49,42,68)(9,48,22,67)(10,47,23,66)(11,46,24,65)(12,45,25,64)(13,44,26,84)(14,43,27,83)(15,63,28,82)(16,62,29,81)(17,61,30,80)(18,60,31,79)(19,59,32,78)(20,58,33,77)(21,57,34,76)(85,145,118,153)(86,144,119,152)(87,143,120,151)(88,142,121,150)(89,141,122,149)(90,140,123,148)(91,139,124,168)(92,138,125,167)(93,137,126,166)(94,136,106,165)(95,135,107,164)(96,134,108,163)(97,133,109,162)(98,132,110,161)(99,131,111,160)(100,130,112,159)(101,129,113,158)(102,128,114,157)(103,127,115,156)(104,147,116,155)(105,146,117,154), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,105)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,168)(54,167)(55,166)(56,165)(57,164)(58,163)(59,162)(60,161)(61,160)(62,159)(63,158)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)>;
G:=Group( (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,144)(44,145)(45,146)(46,147)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,163)(74,164)(75,165)(76,166)(77,167)(78,168)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153), (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), (1,56,35,75)(2,55,36,74)(3,54,37,73)(4,53,38,72)(5,52,39,71)(6,51,40,70)(7,50,41,69)(8,49,42,68)(9,48,22,67)(10,47,23,66)(11,46,24,65)(12,45,25,64)(13,44,26,84)(14,43,27,83)(15,63,28,82)(16,62,29,81)(17,61,30,80)(18,60,31,79)(19,59,32,78)(20,58,33,77)(21,57,34,76)(85,145,118,153)(86,144,119,152)(87,143,120,151)(88,142,121,150)(89,141,122,149)(90,140,123,148)(91,139,124,168)(92,138,125,167)(93,137,126,166)(94,136,106,165)(95,135,107,164)(96,134,108,163)(97,133,109,162)(98,132,110,161)(99,131,111,160)(100,130,112,159)(101,129,113,158)(102,128,114,157)(103,127,115,156)(104,147,116,155)(105,146,117,154), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,105)(12,104)(13,103)(14,102)(15,101)(16,100)(17,99)(18,98)(19,97)(20,96)(21,95)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,168)(54,167)(55,166)(56,165)(57,164)(58,163)(59,162)(60,161)(61,160)(62,159)(63,158)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127) );
G=PermutationGroup([[(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,144),(44,145),(45,146),(46,147),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,154),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,161),(72,162),(73,163),(74,164),(75,165),(76,166),(77,167),(78,168),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153)], [(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)], [(1,56,35,75),(2,55,36,74),(3,54,37,73),(4,53,38,72),(5,52,39,71),(6,51,40,70),(7,50,41,69),(8,49,42,68),(9,48,22,67),(10,47,23,66),(11,46,24,65),(12,45,25,64),(13,44,26,84),(14,43,27,83),(15,63,28,82),(16,62,29,81),(17,61,30,80),(18,60,31,79),(19,59,32,78),(20,58,33,77),(21,57,34,76),(85,145,118,153),(86,144,119,152),(87,143,120,151),(88,142,121,150),(89,141,122,149),(90,140,123,148),(91,139,124,168),(92,138,125,167),(93,137,126,166),(94,136,106,165),(95,135,107,164),(96,134,108,163),(97,133,109,162),(98,132,110,161),(99,131,111,160),(100,130,112,159),(101,129,113,158),(102,128,114,157),(103,127,115,156),(104,147,116,155),(105,146,117,154)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,105),(12,104),(13,103),(14,102),(15,101),(16,100),(17,99),(18,98),(19,97),(20,96),(21,95),(22,119),(23,118),(24,117),(25,116),(26,115),(27,114),(28,113),(29,112),(30,111),(31,110),(32,109),(33,108),(34,107),(35,106),(36,126),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,148),(53,168),(54,167),(55,166),(56,165),(57,164),(58,163),(59,162),(60,161),(61,160),(62,159),(63,158),(64,147),(65,146),(66,145),(67,144),(68,143),(69,142),(70,141),(71,140),(72,139),(73,138),(74,137),(75,136),(76,135),(77,134),(78,133),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | ··· | 6G | 7A | 7B | 7C | 14A | ··· | 14U | 21A | ··· | 21F | 42A | ··· | 42AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | ··· | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 42 | 42 | 2 | 42 | 42 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | C3⋊D4 | D14 | D21 | C7⋊D4 | D42 | C21⋊7D4 |
| kernel | C2×C21⋊7D4 | C2×Dic21 | C21⋊7D4 | C22×D21 | C22×C42 | C22×C14 | C42 | C2×C14 | C22×C6 | C14 | C2×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 9 | 6 | 12 | 18 | 24 |
Matrix representation of C2×C21⋊7D4 ►in GL5(𝔽337)
| 336 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 336 | 336 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 193 | 143 |
| 0 | 0 | 0 | 159 | 34 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 198 | 59 | 0 | 0 |
| 0 | 198 | 139 | 0 | 0 |
| 0 | 0 | 0 | 144 | 336 |
| 0 | 0 | 0 | 178 | 193 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 193 | 1 |
| 0 | 0 | 0 | 159 | 144 |
G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,336,1,0,0,0,336,0,0,0,0,0,0,193,159,0,0,0,143,34],[1,0,0,0,0,0,198,198,0,0,0,59,139,0,0,0,0,0,144,178,0,0,0,336,193],[1,0,0,0,0,0,336,1,0,0,0,0,1,0,0,0,0,0,193,159,0,0,0,1,144] >;
C2×C21⋊7D4 in GAP, Magma, Sage, TeX
C_2\times C_{21}\rtimes_7D_4 % in TeX
G:=Group("C2xC21:7D4"); // GroupNames label
G:=SmallGroup(336,203);
// by ID
G=gap.SmallGroup(336,203);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,964,10373]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations