metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2S3, C8⋊2D21, C24⋊2D7, C168⋊2C2, C4.8D42, C2.3D84, C6.1D28, C21⋊7SD16, D84.1C2, C42.19D4, C28.43D6, 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 C8⋊D21
G = < a,b,c | a24=b7=c2=1, ab=ba, cac=a11, cbc=b-1 >
Smallest permutation representation of C8⋊D21
►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 45 117 152 56 92 144)(2 46 118 153 57 93 121)(3 47 119 154 58 94 122)(4 48 120 155 59 95 123)(5 25 97 156 60 96 124)(6 26 98 157 61 73 125)(7 27 99 158 62 74 126)(8 28 100 159 63 75 127)(9 29 101 160 64 76 128)(10 30 102 161 65 77 129)(11 31 103 162 66 78 130)(12 32 104 163 67 79 131)(13 33 105 164 68 80 132)(14 34 106 165 69 81 133)(15 35 107 166 70 82 134)(16 36 108 167 71 83 135)(17 37 109 168 72 84 136)(18 38 110 145 49 85 137)(19 39 111 146 50 86 138)(20 40 112 147 51 87 139)(21 41 113 148 52 88 140)(22 42 114 149 53 89 141)(23 43 115 150 54 90 142)(24 44 116 151 55 91 143)
(1 144)(2 131)(3 142)(4 129)(5 140)(6 127)(7 138)(8 125)(9 136)(10 123)(11 134)(12 121)(13 132)(14 143)(15 130)(16 141)(17 128)(18 139)(19 126)(20 137)(21 124)(22 135)(23 122)(24 133)(25 88)(26 75)(27 86)(28 73)(29 84)(30 95)(31 82)(32 93)(33 80)(34 91)(35 78)(36 89)(37 76)(38 87)(39 74)(40 85)(41 96)(42 83)(43 94)(44 81)(45 92)(46 79)(47 90)(48 77)(49 112)(50 99)(51 110)(52 97)(53 108)(54 119)(55 106)(56 117)(57 104)(58 115)(59 102)(60 113)(61 100)(62 111)(63 98)(64 109)(65 120)(66 107)(67 118)(68 105)(69 116)(70 103)(71 114)(72 101)(145 147)(146 158)(148 156)(149 167)(150 154)(151 165)(153 163)(155 161)(157 159)(160 168)(162 166)
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), (1,45,117,152,56,92,144)(2,46,118,153,57,93,121)(3,47,119,154,58,94,122)(4,48,120,155,59,95,123)(5,25,97,156,60,96,124)(6,26,98,157,61,73,125)(7,27,99,158,62,74,126)(8,28,100,159,63,75,127)(9,29,101,160,64,76,128)(10,30,102,161,65,77,129)(11,31,103,162,66,78,130)(12,32,104,163,67,79,131)(13,33,105,164,68,80,132)(14,34,106,165,69,81,133)(15,35,107,166,70,82,134)(16,36,108,167,71,83,135)(17,37,109,168,72,84,136)(18,38,110,145,49,85,137)(19,39,111,146,50,86,138)(20,40,112,147,51,87,139)(21,41,113,148,52,88,140)(22,42,114,149,53,89,141)(23,43,115,150,54,90,142)(24,44,116,151,55,91,143), (1,144)(2,131)(3,142)(4,129)(5,140)(6,127)(7,138)(8,125)(9,136)(10,123)(11,134)(12,121)(13,132)(14,143)(15,130)(16,141)(17,128)(18,139)(19,126)(20,137)(21,124)(22,135)(23,122)(24,133)(25,88)(26,75)(27,86)(28,73)(29,84)(30,95)(31,82)(32,93)(33,80)(34,91)(35,78)(36,89)(37,76)(38,87)(39,74)(40,85)(41,96)(42,83)(43,94)(44,81)(45,92)(46,79)(47,90)(48,77)(49,112)(50,99)(51,110)(52,97)(53,108)(54,119)(55,106)(56,117)(57,104)(58,115)(59,102)(60,113)(61,100)(62,111)(63,98)(64,109)(65,120)(66,107)(67,118)(68,105)(69,116)(70,103)(71,114)(72,101)(145,147)(146,158)(148,156)(149,167)(150,154)(151,165)(153,163)(155,161)(157,159)(160,168)(162,166)>;
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), (1,45,117,152,56,92,144)(2,46,118,153,57,93,121)(3,47,119,154,58,94,122)(4,48,120,155,59,95,123)(5,25,97,156,60,96,124)(6,26,98,157,61,73,125)(7,27,99,158,62,74,126)(8,28,100,159,63,75,127)(9,29,101,160,64,76,128)(10,30,102,161,65,77,129)(11,31,103,162,66,78,130)(12,32,104,163,67,79,131)(13,33,105,164,68,80,132)(14,34,106,165,69,81,133)(15,35,107,166,70,82,134)(16,36,108,167,71,83,135)(17,37,109,168,72,84,136)(18,38,110,145,49,85,137)(19,39,111,146,50,86,138)(20,40,112,147,51,87,139)(21,41,113,148,52,88,140)(22,42,114,149,53,89,141)(23,43,115,150,54,90,142)(24,44,116,151,55,91,143), (1,144)(2,131)(3,142)(4,129)(5,140)(6,127)(7,138)(8,125)(9,136)(10,123)(11,134)(12,121)(13,132)(14,143)(15,130)(16,141)(17,128)(18,139)(19,126)(20,137)(21,124)(22,135)(23,122)(24,133)(25,88)(26,75)(27,86)(28,73)(29,84)(30,95)(31,82)(32,93)(33,80)(34,91)(35,78)(36,89)(37,76)(38,87)(39,74)(40,85)(41,96)(42,83)(43,94)(44,81)(45,92)(46,79)(47,90)(48,77)(49,112)(50,99)(51,110)(52,97)(53,108)(54,119)(55,106)(56,117)(57,104)(58,115)(59,102)(60,113)(61,100)(62,111)(63,98)(64,109)(65,120)(66,107)(67,118)(68,105)(69,116)(70,103)(71,114)(72,101)(145,147)(146,158)(148,156)(149,167)(150,154)(151,165)(153,163)(155,161)(157,159)(160,168)(162,166) );
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)], [(1,45,117,152,56,92,144),(2,46,118,153,57,93,121),(3,47,119,154,58,94,122),(4,48,120,155,59,95,123),(5,25,97,156,60,96,124),(6,26,98,157,61,73,125),(7,27,99,158,62,74,126),(8,28,100,159,63,75,127),(9,29,101,160,64,76,128),(10,30,102,161,65,77,129),(11,31,103,162,66,78,130),(12,32,104,163,67,79,131),(13,33,105,164,68,80,132),(14,34,106,165,69,81,133),(15,35,107,166,70,82,134),(16,36,108,167,71,83,135),(17,37,109,168,72,84,136),(18,38,110,145,49,85,137),(19,39,111,146,50,86,138),(20,40,112,147,51,87,139),(21,41,113,148,52,88,140),(22,42,114,149,53,89,141),(23,43,115,150,54,90,142),(24,44,116,151,55,91,143)], [(1,144),(2,131),(3,142),(4,129),(5,140),(6,127),(7,138),(8,125),(9,136),(10,123),(11,134),(12,121),(13,132),(14,143),(15,130),(16,141),(17,128),(18,139),(19,126),(20,137),(21,124),(22,135),(23,122),(24,133),(25,88),(26,75),(27,86),(28,73),(29,84),(30,95),(31,82),(32,93),(33,80),(34,91),(35,78),(36,89),(37,76),(38,87),(39,74),(40,85),(41,96),(42,83),(43,94),(44,81),(45,92),(46,79),(47,90),(48,77),(49,112),(50,99),(51,110),(52,97),(53,108),(54,119),(55,106),(56,117),(57,104),(58,115),(59,102),(60,113),(61,100),(62,111),(63,98),(64,109),(65,120),(66,107),(67,118),(68,105),(69,116),(70,103),(71,114),(72,101),(145,147),(146,158),(148,156),(149,167),(150,154),(151,165),(153,163),(155,161),(157,159),(160,168),(162,166)]])
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 | C8⋊D21 |
| kernel | C8⋊D21 | 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 C8⋊D21 ►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] >;
C8⋊D21 in GAP, Magma, Sage, TeX
C_8\rtimes D_{21} % in TeX
G:=Group("C8:D21"); // 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
Export