direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C2×C12, C84⋊11C22, C42.39C23, C28⋊8(C2×C6), C42⋊5(C2×C4), C14⋊4(C2×C12), (C2×C28)⋊13C6, (C2×C84)⋊10C2, C21⋊6(C22×C4), C7⋊4(C22×C12), Dic7⋊7(C2×C6), D14.9(C2×C6), (C2×C6).37D14, C22.9(C6×D7), (C2×Dic7)⋊11C6, (C6×Dic7)⋊11C2, (C22×D7).4C6, C6.39(C22×D7), C14.16(C22×C6), (C2×C42).38C22, (C6×D7).15C22, (C3×Dic7)⋊10C22, C2.1(C2×C6×D7), (C2×C6×D7).4C2, (C2×C14).26(C2×C6), SmallGroup(336,175)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C2×C12 |
Generators and relations for D7×C2×C12
G = < a,b,c,d | a2=b12=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 336 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C7, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, D7, C14, C14, C22×C4, C21, C2×C12, C2×C12, C22×C6, Dic7, C28, D14, C2×C14, C3×D7, C42, C42, C22×C12, C4×D7, C2×Dic7, C2×C28, C22×D7, C3×Dic7, C84, C6×D7, C2×C42, C2×C4×D7, C12×D7, C6×Dic7, C2×C84, C2×C6×D7, D7×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, D7, C22×C4, C2×C12, C22×C6, D14, C3×D7, C22×C12, C4×D7, C22×D7, C6×D7, C2×C4×D7, C12×D7, C2×C6×D7, D7×C2×C12
►On 168 points
Generators in S168
(1 126)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 121)(9 122)(10 123)(11 124)(12 125)(13 72)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 49)(36 50)(37 108)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 157)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 145)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 133)(117 134)(118 135)(119 136)(120 137)
(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)
(1 115 15 104 161 152 33)(2 116 16 105 162 153 34)(3 117 17 106 163 154 35)(4 118 18 107 164 155 36)(5 119 19 108 165 156 25)(6 120 20 97 166 145 26)(7 109 21 98 167 146 27)(8 110 22 99 168 147 28)(9 111 23 100 157 148 29)(10 112 24 101 158 149 30)(11 113 13 102 159 150 31)(12 114 14 103 160 151 32)(37 80 95 51 130 136 66)(38 81 96 52 131 137 67)(39 82 85 53 132 138 68)(40 83 86 54 121 139 69)(41 84 87 55 122 140 70)(42 73 88 56 123 141 71)(43 74 89 57 124 142 72)(44 75 90 58 125 143 61)(45 76 91 59 126 144 62)(46 77 92 60 127 133 63)(47 78 93 49 128 134 64)(48 79 94 50 129 135 65)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 159)(14 160)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 157)(24 158)(49 128)(50 129)(51 130)(52 131)(53 132)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 133)(93 134)(94 135)(95 136)(96 137)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 145)
G:=sub<Sym(168)| (1,126)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,121)(9,122)(10,123)(11,124)(12,125)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,49)(36,50)(37,108)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,157)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,145)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,133)(117,134)(118,135)(119,136)(120,137), (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), (1,115,15,104,161,152,33)(2,116,16,105,162,153,34)(3,117,17,106,163,154,35)(4,118,18,107,164,155,36)(5,119,19,108,165,156,25)(6,120,20,97,166,145,26)(7,109,21,98,167,146,27)(8,110,22,99,168,147,28)(9,111,23,100,157,148,29)(10,112,24,101,158,149,30)(11,113,13,102,159,150,31)(12,114,14,103,160,151,32)(37,80,95,51,130,136,66)(38,81,96,52,131,137,67)(39,82,85,53,132,138,68)(40,83,86,54,121,139,69)(41,84,87,55,122,140,70)(42,73,88,56,123,141,71)(43,74,89,57,124,142,72)(44,75,90,58,125,143,61)(45,76,91,59,126,144,62)(46,77,92,60,127,133,63)(47,78,93,49,128,134,64)(48,79,94,50,129,135,65), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,157)(24,158)(49,128)(50,129)(51,130)(52,131)(53,132)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,133)(93,134)(94,135)(95,136)(96,137)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,145)>;
G:=Group( (1,126)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,121)(9,122)(10,123)(11,124)(12,125)(13,72)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,49)(36,50)(37,108)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,157)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,145)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,133)(117,134)(118,135)(119,136)(120,137), (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), (1,115,15,104,161,152,33)(2,116,16,105,162,153,34)(3,117,17,106,163,154,35)(4,118,18,107,164,155,36)(5,119,19,108,165,156,25)(6,120,20,97,166,145,26)(7,109,21,98,167,146,27)(8,110,22,99,168,147,28)(9,111,23,100,157,148,29)(10,112,24,101,158,149,30)(11,113,13,102,159,150,31)(12,114,14,103,160,151,32)(37,80,95,51,130,136,66)(38,81,96,52,131,137,67)(39,82,85,53,132,138,68)(40,83,86,54,121,139,69)(41,84,87,55,122,140,70)(42,73,88,56,123,141,71)(43,74,89,57,124,142,72)(44,75,90,58,125,143,61)(45,76,91,59,126,144,62)(46,77,92,60,127,133,63)(47,78,93,49,128,134,64)(48,79,94,50,129,135,65), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,157)(24,158)(49,128)(50,129)(51,130)(52,131)(53,132)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,133)(93,134)(94,135)(95,136)(96,137)(109,146)(110,147)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,145) );
G=PermutationGroup([[(1,126),(2,127),(3,128),(4,129),(5,130),(6,131),(7,132),(8,121),(9,122),(10,123),(11,124),(12,125),(13,72),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,49),(36,50),(37,108),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,157),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,155),(95,156),(96,145),(109,138),(110,139),(111,140),(112,141),(113,142),(114,143),(115,144),(116,133),(117,134),(118,135),(119,136),(120,137)], [(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)], [(1,115,15,104,161,152,33),(2,116,16,105,162,153,34),(3,117,17,106,163,154,35),(4,118,18,107,164,155,36),(5,119,19,108,165,156,25),(6,120,20,97,166,145,26),(7,109,21,98,167,146,27),(8,110,22,99,168,147,28),(9,111,23,100,157,148,29),(10,112,24,101,158,149,30),(11,113,13,102,159,150,31),(12,114,14,103,160,151,32),(37,80,95,51,130,136,66),(38,81,96,52,131,137,67),(39,82,85,53,132,138,68),(40,83,86,54,121,139,69),(41,84,87,55,122,140,70),(42,73,88,56,123,141,71),(43,74,89,57,124,142,72),(44,75,90,58,125,143,61),(45,76,91,59,126,144,62),(46,77,92,60,127,133,63),(47,78,93,49,128,134,64),(48,79,94,50,129,135,65)], [(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,159),(14,160),(15,161),(16,162),(17,163),(18,164),(19,165),(20,166),(21,167),(22,168),(23,157),(24,158),(49,128),(50,129),(51,130),(52,131),(53,132),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(71,73),(72,74),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,144),(92,133),(93,134),(94,135),(95,136),(96,137),(109,146),(110,147),(111,148),(112,149),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,145)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6N | 7A | 7B | 7C | 12A | ··· | 12H | 12I | ··· | 12P | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 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 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D7 | D14 | D14 | C3×D7 | C4×D7 | C6×D7 | C6×D7 | C12×D7 |
| kernel | D7×C2×C12 | C12×D7 | C6×Dic7 | C2×C84 | C2×C6×D7 | C2×C4×D7 | C6×D7 | C4×D7 | C2×Dic7 | C2×C28 | C22×D7 | D14 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 16 | 3 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of D7×C2×C12 ►in GL3(𝔽337) generated by
| 336 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 208 | 0 | 0 |
| 0 | 189 | 0 |
| 0 | 0 | 189 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 336 | 227 |
| 336 | 0 | 0 |
| 0 | 304 | 110 |
| 0 | 33 | 33 |
G:=sub<GL(3,GF(337))| [336,0,0,0,1,0,0,0,1],[208,0,0,0,189,0,0,0,189],[1,0,0,0,0,336,0,1,227],[336,0,0,0,304,33,0,110,33] >;
D7×C2×C12 in GAP, Magma, Sage, TeX
D_7\times C_2\times C_{12} % in TeX
G:=Group("D7xC2xC12"); // GroupNames label
G:=SmallGroup(336,175);
// by ID
G=gap.SmallGroup(336,175);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,122,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations