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G = D168  order 336 = 24·3·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D168, C7⋊1D24, C3⋊1D56, C21⋊4D8, C56⋊1S3, C24⋊1D7, C8⋊1D21, C168⋊1C2, D84⋊1C2, C2.4D84, C4.9D42, C6.2D28, C42.20D4, C28.44D6, C14.2D12, C12.44D14, C84.51C22, sometimes denoted D336 or Dih168 or Dih336, SmallGroup(336,93)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C84 — D168
Chief series
C1 — C7 — C21 — C42 — C84 — D84 — D168
Lower central
C21 — C42 — C84 — D168
Upper central
C1 — C2 — C4 — C8

Generators and relations for D168
 G = < a,b | a168=b2=1, bab=a-1 >

C1
C2
84C2
84C2
C3
C7
C4
42C22
42C22
C6
28S3
28S3
C14
12D7
12D7
C21
C8
21D4
21D4
C12
14D6
14D6
C28
6D14
6D14
C42
4D21
4D21
21D8
C24
7D12
7D12
C56
3D28
3D28
C84
2D42
2D42
7D24
3D56
D84
D84
C168
D168

Smallest permutation representation of D168
►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)

(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 150)(20 149)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)
 
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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)>;
 
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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85) );
 
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)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,156),(14,155),(15,154),(16,153),(17,152),(18,151),(19,150),(20,149),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109),(61,108),(62,107),(63,106),(64,105),(65,104),(66,103),(67,102),(68,101),(69,100),(70,99),(71,98),(72,97),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85)]])
 

87 conjugacy classes

class 1 2A2B2C 3  4  6 7A7B7C8A8B12A12B14A14B14C21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order122234677788121214141421···212424242428···2842···4256···5684···84168···168
size11848422222222222222···222222···22···22···22···22···2

87 irreducible representations

dim11122222222222222
type+++++++++++++++++
imageC1C2C2S3D4D6D7D8D12D14D21D24D28D42D56D84D168
kernelD168C168D84C56C42C28C24C21C14C12C8C7C6C4C3C2C1
# reps11211132236466121224

Matrix representation of D168 ►in GL2(𝔽337) generated by

7125
312276
,
7125
68266
G:=sub<GL(2,GF(337))| [71,312,25,276],[71,68,25,266] >;
 

D168 in GAP, Magma, Sage, TeX 

D_{168}
 
% in TeX
 
G:=Group("D168");
 
// GroupNames label
 
G:=SmallGroup(336,93);
 
// by ID
 
G=gap.SmallGroup(336,93);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,73,79,218,50,964,10373]);
 
// Polycyclic
 
G:=Group<a,b|a^168=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D168 in TeX

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