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G = D84⋊11C2  order 336 = 24·3·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D84⋊11C2, C4.16D42, C28.51D6, C12.47D14, C22.2D42, Dic42⋊11C2, C84.58C22, C42.31C23, D42.5C22, Dic21.8C22, (C2×C28)⋊4S3, (C2×C84)⋊6C2, (C2×C12)⋊4D7, (C2×C4)⋊3D21, (C4×D21)⋊4C2, C3⋊5(C4○D28), C7⋊5(C4○D12), C21⋊7D4⋊7C2, C21⋊11(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

Derived series
C1 — C42 — D84⋊11C2
Chief series
C1 — C7 — C21 — C42 — D42 — C4×D21 — D84⋊11C2
Lower central
C21 — C42 — D84⋊11C2
Upper central
C1 — C4 — C2×C4

Generators and relations for D84⋊11C2
 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, C3, C4, C4, C22, C22, S3, C6, C6, C7, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C2×C12, Dic7, C28, D14, C2×C14, D21, C42, C42, C4○D12, Dic14, C4×D7, D28, C7⋊D4, C2×C28, Dic21, C84, D42, C2×C42, C4○D28, Dic42, C4×D21, D84, C21⋊7D4, C2×C84, D84⋊11C2
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D21, C4○D12, C22×D7, D42, C4○D28, C22×D21, D84⋊11C2

Smallest permutation representation of D84⋊11C2
►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 87)(88 168)(89 167)(90 166)(91 165)(92 164)(93 163)(94 162)(95 161)(96 160)(97 159)(98 158)(99 157)(100 156)(101 155)(102 154)(103 153)(104 152)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)

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

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○D28D84⋊11C2
kernelD84⋊11C2Dic42C4×D21D84C21⋊7D4C2×C84C2×C28C28C2×C14C2×C12C21C12C2×C6C2×C4C7C4C22C3C1
# reps1121211213263641261224

Matrix representation of D84⋊11C2 ►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] >;
 

D84⋊11C2 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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