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G = C2×D84order 336 = 24·3·7

Direct product of C2 and D84

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×D84
 Chief series C1 — C7 — C21 — C42 — D42 — C22×D21 — C2×D84
 Lower central C21 — C42 — C2×D84
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D84
G = < a,b,c | a2=b84=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 896 in 108 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, C23, C12, D6, C2×C6, D7, C14, C14, C2×D4, C21, D12, C2×C12, C22×S3, C28, D14, C2×C14, D21, C42, C42, C2×D12, D28, C2×C28, C22×D7, C84, D42, D42, C2×C42, C2×D28, D84, C2×C84, C22×D21, C2×D84
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, D12, C22×S3, D14, D21, C2×D12, D28, C22×D7, D42, C2×D28, D84, C22×D21, C2×D84

Smallest permutation representation of C2×D84
On 168 points
Generators in S168
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 85)(84 86)
(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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(85 88)(86 87)(89 168)(90 167)(91 166)(92 165)(93 164)(94 163)(95 162)(96 161)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)(105 152)(106 151)(107 150)(108 149)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)(118 139)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 42 42 42 42 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D7 D12 D14 D14 D21 D28 D42 D42 D84 kernel C2×D84 D84 C2×C84 C22×D21 C2×C28 C42 C28 C2×C14 C2×C12 C14 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 2 2 1 3 4 6 3 6 12 12 6 24

Matrix representation of C2×D84 in GL3(𝔽337) generated by

 336 0 0 0 1 0 0 0 1
,
 1 0 0 0 80 235 0 204 39
,
 336 0 0 0 202 122 0 160 135
G:=sub<GL(3,GF(337))| [336,0,0,0,1,0,0,0,1],[1,0,0,0,80,204,0,235,39],[336,0,0,0,202,160,0,122,135] >;

C2×D84 in GAP, Magma, Sage, TeX

C_2\times D_{84}
% in TeX

G:=Group("C2xD84");
// GroupNames label

G:=SmallGroup(336,196);
// by ID

G=gap.SmallGroup(336,196);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,50,964,10373]);
// Polycyclic

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

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