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## G = C2×C7⋊D12order 336 = 24·3·7

### Direct product of C2 and C7⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C7⋊D12
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — C7⋊D12 — C2×C7⋊D12
 Lower central C21 — C42 — C2×C7⋊D12
 Upper central C1 — C22

Generators and relations for C2×C7⋊D12
G = < a,b,c,d | a2=b7=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 668 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, C23, C12, D6, D6, C2×C6, D7, C14, C14, C14, C2×D4, C21, D12, C2×C12, C22×S3, C22×S3, Dic7, D14, C2×C14, C2×C14, S3×C7, D21, C42, C42, C2×D12, C2×Dic7, C7⋊D4, C22×D7, C22×C14, C3×Dic7, S3×C14, S3×C14, D42, D42, C2×C42, C2×C7⋊D4, C7⋊D12, C6×Dic7, S3×C2×C14, C22×D21, C2×C7⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, D12, C22×S3, D14, C2×D12, C7⋊D4, C22×D7, S3×D7, C2×C7⋊D4, C7⋊D12, C2×S3×D7, C2×C7⋊D12

Smallest permutation representation of C2×C7⋊D12
On 168 points
Generators in S168
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 109)(9 110)(10 111)(11 112)(12 113)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 61)(36 62)(37 165)(38 166)(39 167)(40 168)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(81 145)(82 146)(83 147)(84 148)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 121)(95 122)(96 123)(97 141)(98 142)(99 143)(100 144)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)
(1 40 51 35 90 136 83)(2 84 137 91 36 52 41)(3 42 53 25 92 138 73)(4 74 139 93 26 54 43)(5 44 55 27 94 140 75)(6 76 141 95 28 56 45)(7 46 57 29 96 142 77)(8 78 143 85 30 58 47)(9 48 59 31 86 144 79)(10 80 133 87 32 60 37)(11 38 49 33 88 134 81)(12 82 135 89 34 50 39)(13 167 113 146 103 128 72)(14 61 129 104 147 114 168)(15 157 115 148 105 130 62)(16 63 131 106 149 116 158)(17 159 117 150 107 132 64)(18 65 121 108 151 118 160)(19 161 119 152 97 122 66)(20 67 123 98 153 120 162)(21 163 109 154 99 124 68)(22 69 125 100 155 110 164)(23 165 111 156 101 126 70)(24 71 127 102 145 112 166)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 101)(14 100)(15 99)(16 98)(17 97)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)(49 134)(50 133)(51 144)(52 143)(53 142)(54 141)(55 140)(56 139)(57 138)(58 137)(59 136)(60 135)(61 125)(62 124)(63 123)(64 122)(65 121)(66 132)(67 131)(68 130)(69 129)(70 128)(71 127)(72 126)(109 115)(110 114)(111 113)(116 120)(117 119)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 168)(156 167)

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

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

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

54 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 14J ··· 14U 21A 21B 21C 42A ··· 42I order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 14 ··· 14 21 21 21 42 ··· 42 size 1 1 1 1 6 6 42 42 2 14 14 2 2 2 2 2 2 14 14 14 14 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D6 D7 D12 D14 D14 C7⋊D4 S3×D7 C7⋊D12 C2×S3×D7 kernel C2×C7⋊D12 C7⋊D12 C6×Dic7 S3×C2×C14 C22×D21 C2×Dic7 C42 Dic7 C2×C14 C22×S3 C14 D6 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 2 2 1 3 4 6 3 12 3 6 3

Matrix representation of C2×C7⋊D12 in GL5(𝔽337)

 336 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 304 336 0 0 0 305 336
,
 336 0 0 0 0 0 0 336 0 0 0 1 1 0 0 0 0 0 45 215 0 0 0 105 292
,
 336 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 33 1 0 0 0 260 304

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,304,305,0,0,0,336,336],[336,0,0,0,0,0,0,1,0,0,0,336,1,0,0,0,0,0,45,105,0,0,0,215,292],[336,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,33,260,0,0,0,1,304] >;

C2×C7⋊D12 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes D_{12}
% in TeX

G:=Group("C2xC7:D12");
// GroupNames label

G:=SmallGroup(336,159);
// by ID

G=gap.SmallGroup(336,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,121,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^7=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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