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G = D8411C2order 336 = 24·3·7

The semidirect product of D84 and C2 acting through Inn(D84)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8411C2, C4.16D42, C28.51D6, C12.47D14, C22.2D42, Dic4211C2, C84.58C22, C42.31C23, D42.5C22, Dic21.8C22, (C2×C28)⋊4S3, (C2×C84)⋊6C2, (C2×C12)⋊4D7, (C2×C4)⋊3D21, (C4×D21)⋊4C2, C35(C4○D28), C75(C4○D12), C217D47C2, C2111(C4○D4), (C2×C6).29D14, (C2×C14).29D6, C6.31(C22×D7), C2.5(C22×D21), C14.31(C22×S3), (C2×C42).30C22, SmallGroup(336,197)

Series: Derived Chief Lower central Upper central

C1C42 — D8411C2
C1C7C21C42D42C4×D21 — D8411C2
C21C42 — D8411C2
C1C4C2×C4

Generators and relations for D8411C2
 G = < a,b,c | a84=b2=c2=1, bab=a-1, ac=ca, cbc=a42b >

Subgroups: 512 in 80 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C7, C2×C4, C2×C4 [×2], D4 [×3], Q8, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, D7 [×2], C14, C14, C4○D4, C21, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, Dic7 [×2], C28 [×2], D14 [×2], C2×C14, D21 [×2], C42, C42, C4○D12, Dic14, C4×D7 [×2], D28, C7⋊D4 [×2], C2×C28, Dic21 [×2], C84 [×2], D42 [×2], C2×C42, C4○D28, Dic42, C4×D21 [×2], D84, C217D4 [×2], C2×C84, D8411C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], D7, C4○D4, C22×S3, D14 [×3], D21, C4○D12, C22×D7, D42 [×3], C4○D28, C22×D21, D8411C2

Smallest permutation representation of D8411C2
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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(85 91)(86 90)(87 89)(92 168)(93 167)(94 166)(95 165)(96 164)(97 163)(98 162)(99 161)(100 160)(101 159)(102 158)(103 157)(104 156)(105 155)(106 154)(107 153)(108 152)(109 151)(110 150)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)
(1 162)(2 163)(3 164)(4 165)(5 166)(6 167)(7 168)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C7A7B7C12A12B12C12D14A···14I21A···21F28A···28L42A···42R84A···84X
order122223444446667771212121214···1421···2128···2842···4284···84
size11242422112424222222222222···22···22···22···22···2

90 irreducible representations

dim1111112222222222222
type+++++++++++++++
imageC1C2C2C2C2C2S3D6D6D7C4○D4D14D14D21C4○D12D42D42C4○D28D8411C2
kernelD8411C2Dic42C4×D21D84C217D4C2×C84C2×C28C28C2×C14C2×C12C21C12C2×C6C2×C4C7C4C22C3C1
# reps1121211213263641261224

Matrix representation of D8411C2 in GL2(𝔽337) generated by

102158
179122
,
17716
64160
,
12227
310215
G:=sub<GL(2,GF(337))| [102,179,158,122],[177,64,16,160],[122,310,27,215] >;

D8411C2 in GAP, Magma, Sage, TeX

D_{84}\rtimes_{11}C_2
% in TeX

G:=Group("D84:11C2");
// GroupNames label

G:=SmallGroup(336,197);
// by ID

G=gap.SmallGroup(336,197);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,964,10373]);
// Polycyclic

G:=Group<a,b,c|a^84=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^42*b>;
// generators/relations

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