metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊3D7, D6.1D14, C28.15D6, Dic14⋊3S3, C12.16D14, C42.5C23, Dic7.2D6, C84.27C22, D42.9C22, Dic21.11C22, (C4×D21)⋊6C2, (C7×D12)⋊5C2, C21⋊3(C4○D4), C7⋊D12⋊1C2, C4.20(S3×D7), C3⋊1(D4⋊2D7), (S3×Dic7)⋊1C2, C7⋊2(Q8⋊3S3), C6.5(C22×D7), (C3×Dic14)⋊5C2, C14.5(C22×S3), (S3×C14).1C22, (C3×Dic7).2C22, C2.9(C2×S3×D7), SmallGroup(336,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊D7
G = < a,b,c,d | a12=b2=c7=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 436 in 80 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C7, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, D7, C14, C14, C4○D4, C21, C4×S3, D12, D12, C3×Q8, Dic7, Dic7, C28, D14, C2×C14, S3×C7, D21, C42, Q8⋊3S3, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C3×Dic7, Dic21, C84, S3×C14, D42, D4⋊2D7, S3×Dic7, C7⋊D12, C3×Dic14, C7×D12, C4×D21, D12⋊D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, Q8⋊3S3, C22×D7, S3×D7, D4⋊2D7, C2×S3×D7, D12⋊D7
►On 168 points
Generators in S168
(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)
(1 121)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 112)(14 111)(15 110)(16 109)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 142)(26 141)(27 140)(28 139)(29 138)(30 137)(31 136)(32 135)(33 134)(34 133)(35 144)(36 143)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(48 60)(61 85)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 100)(74 99)(75 98)(76 97)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 168)(156 167)
(1 138 115 99 52 167 72)(2 139 116 100 53 168 61)(3 140 117 101 54 157 62)(4 141 118 102 55 158 63)(5 142 119 103 56 159 64)(6 143 120 104 57 160 65)(7 144 109 105 58 161 66)(8 133 110 106 59 162 67)(9 134 111 107 60 163 68)(10 135 112 108 49 164 69)(11 136 113 97 50 165 70)(12 137 114 98 51 166 71)(13 77 47 147 89 124 32)(14 78 48 148 90 125 33)(15 79 37 149 91 126 34)(16 80 38 150 92 127 35)(17 81 39 151 93 128 36)(18 82 40 152 94 129 25)(19 83 41 153 95 130 26)(20 84 42 154 96 131 27)(21 73 43 155 85 132 28)(22 74 44 156 86 121 29)(23 75 45 145 87 122 30)(24 76 46 146 88 123 31)
(1 72)(2 65)(3 70)(4 63)(5 68)(6 61)(7 66)(8 71)(9 64)(10 69)(11 62)(12 67)(13 37)(14 42)(15 47)(16 40)(17 45)(18 38)(19 43)(20 48)(21 41)(22 46)(23 39)(24 44)(25 150)(26 155)(27 148)(28 153)(29 146)(30 151)(31 156)(32 149)(33 154)(34 147)(35 152)(36 145)(49 112)(50 117)(51 110)(52 115)(53 120)(54 113)(55 118)(56 111)(57 116)(58 109)(59 114)(60 119)(73 83)(74 76)(75 81)(77 79)(78 84)(80 82)(85 130)(86 123)(87 128)(88 121)(89 126)(90 131)(91 124)(92 129)(93 122)(94 127)(95 132)(96 125)(97 101)(98 106)(100 104)(103 107)(133 166)(134 159)(135 164)(136 157)(137 162)(138 167)(139 160)(140 165)(141 158)(142 163)(143 168)(144 161)
G:=sub<Sym(168)| (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), (1,121)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,112)(14,111)(15,110)(16,109)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,144)(36,143)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,100)(74,99)(75,98)(76,97)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,168)(156,167), (1,138,115,99,52,167,72)(2,139,116,100,53,168,61)(3,140,117,101,54,157,62)(4,141,118,102,55,158,63)(5,142,119,103,56,159,64)(6,143,120,104,57,160,65)(7,144,109,105,58,161,66)(8,133,110,106,59,162,67)(9,134,111,107,60,163,68)(10,135,112,108,49,164,69)(11,136,113,97,50,165,70)(12,137,114,98,51,166,71)(13,77,47,147,89,124,32)(14,78,48,148,90,125,33)(15,79,37,149,91,126,34)(16,80,38,150,92,127,35)(17,81,39,151,93,128,36)(18,82,40,152,94,129,25)(19,83,41,153,95,130,26)(20,84,42,154,96,131,27)(21,73,43,155,85,132,28)(22,74,44,156,86,121,29)(23,75,45,145,87,122,30)(24,76,46,146,88,123,31), (1,72)(2,65)(3,70)(4,63)(5,68)(6,61)(7,66)(8,71)(9,64)(10,69)(11,62)(12,67)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44)(25,150)(26,155)(27,148)(28,153)(29,146)(30,151)(31,156)(32,149)(33,154)(34,147)(35,152)(36,145)(49,112)(50,117)(51,110)(52,115)(53,120)(54,113)(55,118)(56,111)(57,116)(58,109)(59,114)(60,119)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82)(85,130)(86,123)(87,128)(88,121)(89,126)(90,131)(91,124)(92,129)(93,122)(94,127)(95,132)(96,125)(97,101)(98,106)(100,104)(103,107)(133,166)(134,159)(135,164)(136,157)(137,162)(138,167)(139,160)(140,165)(141,158)(142,163)(143,168)(144,161)>;
G:=Group( (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), (1,121)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,112)(14,111)(15,110)(16,109)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,144)(36,143)(37,59)(38,58)(39,57)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(48,60)(61,85)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,100)(74,99)(75,98)(76,97)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,168)(156,167), (1,138,115,99,52,167,72)(2,139,116,100,53,168,61)(3,140,117,101,54,157,62)(4,141,118,102,55,158,63)(5,142,119,103,56,159,64)(6,143,120,104,57,160,65)(7,144,109,105,58,161,66)(8,133,110,106,59,162,67)(9,134,111,107,60,163,68)(10,135,112,108,49,164,69)(11,136,113,97,50,165,70)(12,137,114,98,51,166,71)(13,77,47,147,89,124,32)(14,78,48,148,90,125,33)(15,79,37,149,91,126,34)(16,80,38,150,92,127,35)(17,81,39,151,93,128,36)(18,82,40,152,94,129,25)(19,83,41,153,95,130,26)(20,84,42,154,96,131,27)(21,73,43,155,85,132,28)(22,74,44,156,86,121,29)(23,75,45,145,87,122,30)(24,76,46,146,88,123,31), (1,72)(2,65)(3,70)(4,63)(5,68)(6,61)(7,66)(8,71)(9,64)(10,69)(11,62)(12,67)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44)(25,150)(26,155)(27,148)(28,153)(29,146)(30,151)(31,156)(32,149)(33,154)(34,147)(35,152)(36,145)(49,112)(50,117)(51,110)(52,115)(53,120)(54,113)(55,118)(56,111)(57,116)(58,109)(59,114)(60,119)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82)(85,130)(86,123)(87,128)(88,121)(89,126)(90,131)(91,124)(92,129)(93,122)(94,127)(95,132)(96,125)(97,101)(98,106)(100,104)(103,107)(133,166)(134,159)(135,164)(136,157)(137,162)(138,167)(139,160)(140,165)(141,158)(142,163)(143,168)(144,161) );
G=PermutationGroup([[(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)], [(1,121),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,112),(14,111),(15,110),(16,109),(17,120),(18,119),(19,118),(20,117),(21,116),(22,115),(23,114),(24,113),(25,142),(26,141),(27,140),(28,139),(29,138),(30,137),(31,136),(32,135),(33,134),(34,133),(35,144),(36,143),(37,59),(38,58),(39,57),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(48,60),(61,85),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86),(73,100),(74,99),(75,98),(76,97),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101),(145,166),(146,165),(147,164),(148,163),(149,162),(150,161),(151,160),(152,159),(153,158),(154,157),(155,168),(156,167)], [(1,138,115,99,52,167,72),(2,139,116,100,53,168,61),(3,140,117,101,54,157,62),(4,141,118,102,55,158,63),(5,142,119,103,56,159,64),(6,143,120,104,57,160,65),(7,144,109,105,58,161,66),(8,133,110,106,59,162,67),(9,134,111,107,60,163,68),(10,135,112,108,49,164,69),(11,136,113,97,50,165,70),(12,137,114,98,51,166,71),(13,77,47,147,89,124,32),(14,78,48,148,90,125,33),(15,79,37,149,91,126,34),(16,80,38,150,92,127,35),(17,81,39,151,93,128,36),(18,82,40,152,94,129,25),(19,83,41,153,95,130,26),(20,84,42,154,96,131,27),(21,73,43,155,85,132,28),(22,74,44,156,86,121,29),(23,75,45,145,87,122,30),(24,76,46,146,88,123,31)], [(1,72),(2,65),(3,70),(4,63),(5,68),(6,61),(7,66),(8,71),(9,64),(10,69),(11,62),(12,67),(13,37),(14,42),(15,47),(16,40),(17,45),(18,38),(19,43),(20,48),(21,41),(22,46),(23,39),(24,44),(25,150),(26,155),(27,148),(28,153),(29,146),(30,151),(31,156),(32,149),(33,154),(34,147),(35,152),(36,145),(49,112),(50,117),(51,110),(52,115),(53,120),(54,113),(55,118),(56,111),(57,116),(58,109),(59,114),(60,119),(73,83),(74,76),(75,81),(77,79),(78,84),(80,82),(85,130),(86,123),(87,128),(88,121),(89,126),(90,131),(91,124),(92,129),(93,122),(94,127),(95,132),(96,125),(97,101),(98,106),(100,104),(103,107),(133,166),(134,159),(135,164),(136,157),(137,162),(138,167),(139,160),(140,165),(141,158),(142,163),(143,168),(144,161)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 7A | 7B | 7C | 12A | 12B | 12C | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 6 | 42 | 2 | 2 | 14 | 14 | 21 | 21 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D7 | C4○D4 | D14 | D14 | Q8⋊3S3 | S3×D7 | D4⋊2D7 | C2×S3×D7 | D12⋊D7 |
| kernel | D12⋊D7 | S3×Dic7 | C7⋊D12 | C3×Dic14 | C7×D12 | C4×D21 | Dic14 | Dic7 | C28 | D12 | C21 | C12 | D6 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 6 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of D12⋊D7 ►in GL6(𝔽337)
| 189 | 0 | 0 | 0 | 0 | 0 |
| 266 | 148 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 141 |
| 0 | 0 | 0 | 0 | 43 | 335 |
| 68 | 63 | 0 | 0 | 0 | 0 |
| 55 | 269 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 141 |
| 0 | 0 | 0 | 0 | 43 | 335 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 1 | 0 | 0 |
| 0 | 0 | 32 | 304 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 276 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 32 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 196 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(337))| [189,266,0,0,0,0,0,148,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,43,0,0,0,0,141,335],[68,55,0,0,0,0,63,269,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,43,0,0,0,0,141,335],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,32,0,0,0,0,1,304,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,276,0,0,0,0,0,336,0,0,0,0,0,0,336,32,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,196,1] >;
D12⋊D7 in GAP, Magma, Sage, TeX
D_{12}\rtimes D_7 % in TeX
G:=Group("D12:D7"); // GroupNames label
G:=SmallGroup(336,141);
// by ID
G=gap.SmallGroup(336,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,218,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations