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G = C2×C7⋊D12order 336 = 24·3·7

Direct product of C2 and C7⋊D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C7⋊D12, C423D4, D66D14, C142D12, Dic74D6, D429C22, C42.23C23, C73(C2×D12), C216(C2×D4), C61(C7⋊D4), (C6×Dic7)⋊6C2, (C2×Dic7)⋊4S3, (C2×C14).18D6, (C2×C6).18D14, (C22×S3)⋊2D7, (S3×C14)⋊6C22, (C22×D21)⋊5C2, C22.16(S3×D7), C6.23(C22×D7), C14.23(C22×S3), (C2×C42).17C22, (C3×Dic7)⋊7C22, C31(C2×C7⋊D4), (S3×C2×C14)⋊2C2, C2.23(C2×S3×D7), SmallGroup(336,159)

Series: Derived Chief Lower central Upper central

C1C42 — C2×C7⋊D12
C1C7C21C42C3×Dic7C7⋊D12 — C2×C7⋊D12
C21C42 — C2×C7⋊D12
C1C22

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 2A2B2C2D2E2F2G 3 4A4B6A6B6C7A7B7C12A12B12C12D14A···14I14J···14U21A21B21C42A···42I
order122222223446667771212121214···1414···1421212142···42
size111166424221414222222141414142···26···64444···4

54 irreducible representations

dim11111222222222444
type++++++++++++++++
imageC1C2C2C2C2S3D4D6D6D7D12D14D14C7⋊D4S3×D7C7⋊D12C2×S3×D7
kernelC2×C7⋊D12C7⋊D12C6×Dic7S3×C2×C14C22×D21C2×Dic7C42Dic7C2×C14C22×S3C14D6C2×C6C6C22C2C2
# reps141111221346312363

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

3360000
01000
00100
00010
00001
,
10000
01000
00100
000304336
000305336
,
3360000
0033600
01100
00045215
000105292
,
3360000
00100
01000
000331
000260304

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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