direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C21, D4⋊C42, C8⋊1C42, C56⋊13C6, C24⋊3C14, C168⋊11C2, C42.54D4, C84.77C22, (C7×D4)⋊10C6, (C3×D4)⋊4C14, C4.1(C2×C42), C2.3(D4×C21), C6.14(C7×D4), (D4×C21)⋊10C2, C28.40(C2×C6), C14.30(C3×D4), C12.17(C2×C14), SmallGroup(336,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C21
G = < a,b,c | a21=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C21
►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 41 77 134 125 148 97 57)(2 42 78 135 126 149 98 58)(3 22 79 136 106 150 99 59)(4 23 80 137 107 151 100 60)(5 24 81 138 108 152 101 61)(6 25 82 139 109 153 102 62)(7 26 83 140 110 154 103 63)(8 27 84 141 111 155 104 43)(9 28 64 142 112 156 105 44)(10 29 65 143 113 157 85 45)(11 30 66 144 114 158 86 46)(12 31 67 145 115 159 87 47)(13 32 68 146 116 160 88 48)(14 33 69 147 117 161 89 49)(15 34 70 127 118 162 90 50)(16 35 71 128 119 163 91 51)(17 36 72 129 120 164 92 52)(18 37 73 130 121 165 93 53)(19 38 74 131 122 166 94 54)(20 39 75 132 123 167 95 55)(21 40 76 133 124 168 96 56)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 127)(119 128)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)
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), (1,41,77,134,125,148,97,57)(2,42,78,135,126,149,98,58)(3,22,79,136,106,150,99,59)(4,23,80,137,107,151,100,60)(5,24,81,138,108,152,101,61)(6,25,82,139,109,153,102,62)(7,26,83,140,110,154,103,63)(8,27,84,141,111,155,104,43)(9,28,64,142,112,156,105,44)(10,29,65,143,113,157,85,45)(11,30,66,144,114,158,86,46)(12,31,67,145,115,159,87,47)(13,32,68,146,116,160,88,48)(14,33,69,147,117,161,89,49)(15,34,70,127,118,162,90,50)(16,35,71,128,119,163,91,51)(17,36,72,129,120,164,92,52)(18,37,73,130,121,165,93,53)(19,38,74,131,122,166,94,54)(20,39,75,132,123,167,95,55)(21,40,76,133,124,168,96,56), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,127)(119,128)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)>;
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), (1,41,77,134,125,148,97,57)(2,42,78,135,126,149,98,58)(3,22,79,136,106,150,99,59)(4,23,80,137,107,151,100,60)(5,24,81,138,108,152,101,61)(6,25,82,139,109,153,102,62)(7,26,83,140,110,154,103,63)(8,27,84,141,111,155,104,43)(9,28,64,142,112,156,105,44)(10,29,65,143,113,157,85,45)(11,30,66,144,114,158,86,46)(12,31,67,145,115,159,87,47)(13,32,68,146,116,160,88,48)(14,33,69,147,117,161,89,49)(15,34,70,127,118,162,90,50)(16,35,71,128,119,163,91,51)(17,36,72,129,120,164,92,52)(18,37,73,130,121,165,93,53)(19,38,74,131,122,166,94,54)(20,39,75,132,123,167,95,55)(21,40,76,133,124,168,96,56), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,127)(119,128)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135) );
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)], [(1,41,77,134,125,148,97,57),(2,42,78,135,126,149,98,58),(3,22,79,136,106,150,99,59),(4,23,80,137,107,151,100,60),(5,24,81,138,108,152,101,61),(6,25,82,139,109,153,102,62),(7,26,83,140,110,154,103,63),(8,27,84,141,111,155,104,43),(9,28,64,142,112,156,105,44),(10,29,65,143,113,157,85,45),(11,30,66,144,114,158,86,46),(12,31,67,145,115,159,87,47),(13,32,68,146,116,160,88,48),(14,33,69,147,117,161,89,49),(15,34,70,127,118,162,90,50),(16,35,71,128,119,163,91,51),(17,36,72,129,120,164,92,52),(18,37,73,130,121,165,93,53),(19,38,74,131,122,166,94,54),(20,39,75,132,123,167,95,55),(21,40,76,133,124,168,96,56)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(106,136),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144),(115,145),(116,146),(117,147),(118,127),(119,128),(120,129),(121,130),(122,131),(123,132),(124,133),(125,134),(126,135)]])
147 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | ··· | 7F | 8A | 8B | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21L | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42L | 42M | ··· | 42AJ | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
147 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C7 | C14 | C14 | C21 | C42 | C42 | D4 | D8 | C3×D4 | C3×D8 | C7×D4 | C7×D8 | D4×C21 | D8×C21 |
| kernel | D8×C21 | C168 | D4×C21 | C7×D8 | C56 | C7×D4 | C3×D8 | C24 | C3×D4 | D8 | C8 | D4 | C42 | C21 | C14 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 12 | 24 | 1 | 2 | 2 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of D8×C21 ►in GL2(𝔽337) generated by
| 4 | 0 |
| 0 | 4 |
| 324 | 13 |
| 324 | 324 |
| 324 | 13 |
| 13 | 13 |
G:=sub<GL(2,GF(337))| [4,0,0,4],[324,324,13,324],[324,13,13,13] >;
D8×C21 in GAP, Magma, Sage, TeX
D_8\times C_{21} % in TeX
G:=Group("D8xC21"); // GroupNames label
G:=SmallGroup(336,111);
// by ID
G=gap.SmallGroup(336,111);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1033,7564,3790,88]);
// Polycyclic
G:=Group<a,b,c|a^21=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export