direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C7⋊D12, C42⋊3D4, D6⋊6D14, C14⋊2D12, Dic7⋊4D6, D42⋊9C22, C42.23C23, C7⋊3(C2×D12), C21⋊6(C2×D4), C6⋊1(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, C3⋊1(C2×C7⋊D4), (S3×C2×C14)⋊2C2, C2.23(C2×S3×D7), SmallGroup(336,159)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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