metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2S3, C24⋊2D7, C8⋊2D21, C168⋊2C2, C2.3D84, C4.8D42, C6.1D28, C21⋊7SD16, D84.1C2, C28.43D6, C42.19D4, C14.1D12, Dic42⋊1C2, C12.43D14, C84.50C22, C7⋊1(C24⋊C2), C3⋊1(C56⋊C2), SmallGroup(336,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D7
G = < a,b,c | a24=b7=c2=1, ab=ba, cac=a11, cbc=b-1 >
Smallest permutation representation of C24⋊D7
►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 34 66 97 168 134 92)(2 35 67 98 145 135 93)(3 36 68 99 146 136 94)(4 37 69 100 147 137 95)(5 38 70 101 148 138 96)(6 39 71 102 149 139 73)(7 40 72 103 150 140 74)(8 41 49 104 151 141 75)(9 42 50 105 152 142 76)(10 43 51 106 153 143 77)(11 44 52 107 154 144 78)(12 45 53 108 155 121 79)(13 46 54 109 156 122 80)(14 47 55 110 157 123 81)(15 48 56 111 158 124 82)(16 25 57 112 159 125 83)(17 26 58 113 160 126 84)(18 27 59 114 161 127 85)(19 28 60 115 162 128 86)(20 29 61 116 163 129 87)(21 30 62 117 164 130 88)(22 31 63 118 165 131 89)(23 32 64 119 166 132 90)(24 33 65 120 167 133 91)
(1 92)(2 79)(3 90)(4 77)(5 88)(6 75)(7 86)(8 73)(9 84)(10 95)(11 82)(12 93)(13 80)(14 91)(15 78)(16 89)(17 76)(18 87)(19 74)(20 85)(21 96)(22 83)(23 94)(24 81)(25 131)(26 142)(27 129)(28 140)(29 127)(30 138)(31 125)(32 136)(33 123)(34 134)(35 121)(36 132)(37 143)(38 130)(39 141)(40 128)(41 139)(42 126)(43 137)(44 124)(45 135)(46 122)(47 133)(48 144)(49 149)(50 160)(51 147)(52 158)(53 145)(54 156)(55 167)(56 154)(57 165)(58 152)(59 163)(60 150)(61 161)(62 148)(63 159)(64 146)(65 157)(66 168)(67 155)(68 166)(69 153)(70 164)(71 151)(72 162)(98 108)(99 119)(100 106)(101 117)(102 104)(103 115)(105 113)(107 111)(110 120)(112 118)(114 116)
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), (1,34,66,97,168,134,92)(2,35,67,98,145,135,93)(3,36,68,99,146,136,94)(4,37,69,100,147,137,95)(5,38,70,101,148,138,96)(6,39,71,102,149,139,73)(7,40,72,103,150,140,74)(8,41,49,104,151,141,75)(9,42,50,105,152,142,76)(10,43,51,106,153,143,77)(11,44,52,107,154,144,78)(12,45,53,108,155,121,79)(13,46,54,109,156,122,80)(14,47,55,110,157,123,81)(15,48,56,111,158,124,82)(16,25,57,112,159,125,83)(17,26,58,113,160,126,84)(18,27,59,114,161,127,85)(19,28,60,115,162,128,86)(20,29,61,116,163,129,87)(21,30,62,117,164,130,88)(22,31,63,118,165,131,89)(23,32,64,119,166,132,90)(24,33,65,120,167,133,91), (1,92)(2,79)(3,90)(4,77)(5,88)(6,75)(7,86)(8,73)(9,84)(10,95)(11,82)(12,93)(13,80)(14,91)(15,78)(16,89)(17,76)(18,87)(19,74)(20,85)(21,96)(22,83)(23,94)(24,81)(25,131)(26,142)(27,129)(28,140)(29,127)(30,138)(31,125)(32,136)(33,123)(34,134)(35,121)(36,132)(37,143)(38,130)(39,141)(40,128)(41,139)(42,126)(43,137)(44,124)(45,135)(46,122)(47,133)(48,144)(49,149)(50,160)(51,147)(52,158)(53,145)(54,156)(55,167)(56,154)(57,165)(58,152)(59,163)(60,150)(61,161)(62,148)(63,159)(64,146)(65,157)(66,168)(67,155)(68,166)(69,153)(70,164)(71,151)(72,162)(98,108)(99,119)(100,106)(101,117)(102,104)(103,115)(105,113)(107,111)(110,120)(112,118)(114,116)>;
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), (1,34,66,97,168,134,92)(2,35,67,98,145,135,93)(3,36,68,99,146,136,94)(4,37,69,100,147,137,95)(5,38,70,101,148,138,96)(6,39,71,102,149,139,73)(7,40,72,103,150,140,74)(8,41,49,104,151,141,75)(9,42,50,105,152,142,76)(10,43,51,106,153,143,77)(11,44,52,107,154,144,78)(12,45,53,108,155,121,79)(13,46,54,109,156,122,80)(14,47,55,110,157,123,81)(15,48,56,111,158,124,82)(16,25,57,112,159,125,83)(17,26,58,113,160,126,84)(18,27,59,114,161,127,85)(19,28,60,115,162,128,86)(20,29,61,116,163,129,87)(21,30,62,117,164,130,88)(22,31,63,118,165,131,89)(23,32,64,119,166,132,90)(24,33,65,120,167,133,91), (1,92)(2,79)(3,90)(4,77)(5,88)(6,75)(7,86)(8,73)(9,84)(10,95)(11,82)(12,93)(13,80)(14,91)(15,78)(16,89)(17,76)(18,87)(19,74)(20,85)(21,96)(22,83)(23,94)(24,81)(25,131)(26,142)(27,129)(28,140)(29,127)(30,138)(31,125)(32,136)(33,123)(34,134)(35,121)(36,132)(37,143)(38,130)(39,141)(40,128)(41,139)(42,126)(43,137)(44,124)(45,135)(46,122)(47,133)(48,144)(49,149)(50,160)(51,147)(52,158)(53,145)(54,156)(55,167)(56,154)(57,165)(58,152)(59,163)(60,150)(61,161)(62,148)(63,159)(64,146)(65,157)(66,168)(67,155)(68,166)(69,153)(70,164)(71,151)(72,162)(98,108)(99,119)(100,106)(101,117)(102,104)(103,115)(105,113)(107,111)(110,120)(112,118)(114,116) );
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)], [(1,34,66,97,168,134,92),(2,35,67,98,145,135,93),(3,36,68,99,146,136,94),(4,37,69,100,147,137,95),(5,38,70,101,148,138,96),(6,39,71,102,149,139,73),(7,40,72,103,150,140,74),(8,41,49,104,151,141,75),(9,42,50,105,152,142,76),(10,43,51,106,153,143,77),(11,44,52,107,154,144,78),(12,45,53,108,155,121,79),(13,46,54,109,156,122,80),(14,47,55,110,157,123,81),(15,48,56,111,158,124,82),(16,25,57,112,159,125,83),(17,26,58,113,160,126,84),(18,27,59,114,161,127,85),(19,28,60,115,162,128,86),(20,29,61,116,163,129,87),(21,30,62,117,164,130,88),(22,31,63,118,165,131,89),(23,32,64,119,166,132,90),(24,33,65,120,167,133,91)], [(1,92),(2,79),(3,90),(4,77),(5,88),(6,75),(7,86),(8,73),(9,84),(10,95),(11,82),(12,93),(13,80),(14,91),(15,78),(16,89),(17,76),(18,87),(19,74),(20,85),(21,96),(22,83),(23,94),(24,81),(25,131),(26,142),(27,129),(28,140),(29,127),(30,138),(31,125),(32,136),(33,123),(34,134),(35,121),(36,132),(37,143),(38,130),(39,141),(40,128),(41,139),(42,126),(43,137),(44,124),(45,135),(46,122),(47,133),(48,144),(49,149),(50,160),(51,147),(52,158),(53,145),(54,156),(55,167),(56,154),(57,165),(58,152),(59,163),(60,150),(61,161),(62,148),(63,159),(64,146),(65,157),(66,168),(67,155),(68,166),(69,153),(70,164),(71,151),(72,162),(98,108),(99,119),(100,106),(101,117),(102,104),(103,115),(105,113),(107,111),(110,120),(112,118),(114,116)])
87 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 84 | 2 | 2 | 84 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | SD16 | D12 | D14 | D21 | C24⋊C2 | D28 | D42 | C56⋊C2 | D84 | C24⋊D7 |
| kernel | C24⋊D7 | C168 | Dic42 | D84 | C56 | C42 | C28 | C24 | C21 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C24⋊D7 ►in GL2(𝔽337) generated by
| 158 | 209 |
| 128 | 52 |
| 0 | 1 |
| 336 | 143 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(337))| [158,128,209,52],[0,336,1,143],[0,1,1,0] >;
C24⋊D7 in GAP, Magma, Sage, TeX
C_{24}\rtimes D_7 % in TeX
G:=Group("C24:D7"); // GroupNames label
G:=SmallGroup(336,92);
// by ID
G=gap.SmallGroup(336,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,31,218,50,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^24=b^7=c^2=1,a*b=b*a,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations