metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.8D4, C28.5D6, C21⋊5SD16, D12.1D7, C14.7D12, Dic42⋊8C2, C12.23D14, C84.16C22, C7⋊C8⋊2S3, C4.9(S3×D7), C7⋊3(C24⋊C2), C3⋊1(D4.D7), (C7×D12).1C2, C6.2(C7⋊D4), C2.5(C7⋊D12), (C3×C7⋊C8)⋊2C2, SmallGroup(336,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.D7
G = < a,b,c,d | a12=b2=c7=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Smallest permutation representation of 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 53)(50 52)(54 60)(55 59)(56 58)(61 69)(62 68)(63 67)(64 66)(70 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 93)(86 92)(87 91)(88 90)(94 96)(97 101)(98 100)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 129)(122 128)(123 127)(124 126)(130 132)(133 135)(136 144)(137 143)(138 142)(139 141)(145 148)(146 147)(149 156)(150 155)(151 154)(152 153)(157 164)(158 163)(159 162)(160 161)(165 168)(166 167)
(1 79 30 13 45 153 161)(2 80 31 14 46 154 162)(3 81 32 15 47 155 163)(4 82 33 16 48 156 164)(5 83 34 17 37 145 165)(6 84 35 18 38 146 166)(7 73 36 19 39 147 167)(8 74 25 20 40 148 168)(9 75 26 21 41 149 157)(10 76 27 22 42 150 158)(11 77 28 23 43 151 159)(12 78 29 24 44 152 160)(49 119 129 138 87 103 63)(50 120 130 139 88 104 64)(51 109 131 140 89 105 65)(52 110 132 141 90 106 66)(53 111 121 142 91 107 67)(54 112 122 143 92 108 68)(55 113 123 144 93 97 69)(56 114 124 133 94 98 70)(57 115 125 134 95 99 71)(58 116 126 135 96 100 72)(59 117 127 136 85 101 61)(60 118 128 137 86 102 62)
(1 107 10 104 7 101 4 98)(2 108 11 105 8 102 5 99)(3 97 12 106 9 103 6 100)(13 121 22 130 19 127 16 124)(14 122 23 131 20 128 17 125)(15 123 24 132 21 129 18 126)(25 137 34 134 31 143 28 140)(26 138 35 135 32 144 29 141)(27 139 36 136 33 133 30 142)(37 115 46 112 43 109 40 118)(38 116 47 113 44 110 41 119)(39 117 48 114 45 111 42 120)(49 146 58 155 55 152 52 149)(50 147 59 156 56 153 53 150)(51 148 60 145 57 154 54 151)(61 164 70 161 67 158 64 167)(62 165 71 162 68 159 65 168)(63 166 72 163 69 160 66 157)(73 85 82 94 79 91 76 88)(74 86 83 95 80 92 77 89)(75 87 84 96 81 93 78 90)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,53)(50,52)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(70,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,93)(86,92)(87,91)(88,90)(94,96)(97,101)(98,100)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(130,132)(133,135)(136,144)(137,143)(138,142)(139,141)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,164)(158,163)(159,162)(160,161)(165,168)(166,167), (1,79,30,13,45,153,161)(2,80,31,14,46,154,162)(3,81,32,15,47,155,163)(4,82,33,16,48,156,164)(5,83,34,17,37,145,165)(6,84,35,18,38,146,166)(7,73,36,19,39,147,167)(8,74,25,20,40,148,168)(9,75,26,21,41,149,157)(10,76,27,22,42,150,158)(11,77,28,23,43,151,159)(12,78,29,24,44,152,160)(49,119,129,138,87,103,63)(50,120,130,139,88,104,64)(51,109,131,140,89,105,65)(52,110,132,141,90,106,66)(53,111,121,142,91,107,67)(54,112,122,143,92,108,68)(55,113,123,144,93,97,69)(56,114,124,133,94,98,70)(57,115,125,134,95,99,71)(58,116,126,135,96,100,72)(59,117,127,136,85,101,61)(60,118,128,137,86,102,62), (1,107,10,104,7,101,4,98)(2,108,11,105,8,102,5,99)(3,97,12,106,9,103,6,100)(13,121,22,130,19,127,16,124)(14,122,23,131,20,128,17,125)(15,123,24,132,21,129,18,126)(25,137,34,134,31,143,28,140)(26,138,35,135,32,144,29,141)(27,139,36,136,33,133,30,142)(37,115,46,112,43,109,40,118)(38,116,47,113,44,110,41,119)(39,117,48,114,45,111,42,120)(49,146,58,155,55,152,52,149)(50,147,59,156,56,153,53,150)(51,148,60,145,57,154,54,151)(61,164,70,161,67,158,64,167)(62,165,71,162,68,159,65,168)(63,166,72,163,69,160,66,157)(73,85,82,94,79,91,76,88)(74,86,83,95,80,92,77,89)(75,87,84,96,81,93,78,90)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,53)(50,52)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(70,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,93)(86,92)(87,91)(88,90)(94,96)(97,101)(98,100)(102,108)(103,107)(104,106)(110,120)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(130,132)(133,135)(136,144)(137,143)(138,142)(139,141)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,164)(158,163)(159,162)(160,161)(165,168)(166,167), (1,79,30,13,45,153,161)(2,80,31,14,46,154,162)(3,81,32,15,47,155,163)(4,82,33,16,48,156,164)(5,83,34,17,37,145,165)(6,84,35,18,38,146,166)(7,73,36,19,39,147,167)(8,74,25,20,40,148,168)(9,75,26,21,41,149,157)(10,76,27,22,42,150,158)(11,77,28,23,43,151,159)(12,78,29,24,44,152,160)(49,119,129,138,87,103,63)(50,120,130,139,88,104,64)(51,109,131,140,89,105,65)(52,110,132,141,90,106,66)(53,111,121,142,91,107,67)(54,112,122,143,92,108,68)(55,113,123,144,93,97,69)(56,114,124,133,94,98,70)(57,115,125,134,95,99,71)(58,116,126,135,96,100,72)(59,117,127,136,85,101,61)(60,118,128,137,86,102,62), (1,107,10,104,7,101,4,98)(2,108,11,105,8,102,5,99)(3,97,12,106,9,103,6,100)(13,121,22,130,19,127,16,124)(14,122,23,131,20,128,17,125)(15,123,24,132,21,129,18,126)(25,137,34,134,31,143,28,140)(26,138,35,135,32,144,29,141)(27,139,36,136,33,133,30,142)(37,115,46,112,43,109,40,118)(38,116,47,113,44,110,41,119)(39,117,48,114,45,111,42,120)(49,146,58,155,55,152,52,149)(50,147,59,156,56,153,53,150)(51,148,60,145,57,154,54,151)(61,164,70,161,67,158,64,167)(62,165,71,162,68,159,65,168)(63,166,72,163,69,160,66,157)(73,85,82,94,79,91,76,88)(74,86,83,95,80,92,77,89)(75,87,84,96,81,93,78,90) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,53),(50,52),(54,60),(55,59),(56,58),(61,69),(62,68),(63,67),(64,66),(70,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,93),(86,92),(87,91),(88,90),(94,96),(97,101),(98,100),(102,108),(103,107),(104,106),(110,120),(111,119),(112,118),(113,117),(114,116),(121,129),(122,128),(123,127),(124,126),(130,132),(133,135),(136,144),(137,143),(138,142),(139,141),(145,148),(146,147),(149,156),(150,155),(151,154),(152,153),(157,164),(158,163),(159,162),(160,161),(165,168),(166,167)], [(1,79,30,13,45,153,161),(2,80,31,14,46,154,162),(3,81,32,15,47,155,163),(4,82,33,16,48,156,164),(5,83,34,17,37,145,165),(6,84,35,18,38,146,166),(7,73,36,19,39,147,167),(8,74,25,20,40,148,168),(9,75,26,21,41,149,157),(10,76,27,22,42,150,158),(11,77,28,23,43,151,159),(12,78,29,24,44,152,160),(49,119,129,138,87,103,63),(50,120,130,139,88,104,64),(51,109,131,140,89,105,65),(52,110,132,141,90,106,66),(53,111,121,142,91,107,67),(54,112,122,143,92,108,68),(55,113,123,144,93,97,69),(56,114,124,133,94,98,70),(57,115,125,134,95,99,71),(58,116,126,135,96,100,72),(59,117,127,136,85,101,61),(60,118,128,137,86,102,62)], [(1,107,10,104,7,101,4,98),(2,108,11,105,8,102,5,99),(3,97,12,106,9,103,6,100),(13,121,22,130,19,127,16,124),(14,122,23,131,20,128,17,125),(15,123,24,132,21,129,18,126),(25,137,34,134,31,143,28,140),(26,138,35,135,32,144,29,141),(27,139,36,136,33,133,30,142),(37,115,46,112,43,109,40,118),(38,116,47,113,44,110,41,119),(39,117,48,114,45,111,42,120),(49,146,58,155,55,152,52,149),(50,147,59,156,56,153,53,150),(51,148,60,145,57,154,54,151),(61,164,70,161,67,158,64,167),(62,165,71,162,68,159,65,168),(63,166,72,163,69,160,66,157),(73,85,82,94,79,91,76,88),(74,86,83,95,80,92,77,89),(75,87,84,96,81,93,78,90)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 24A | 24B | 24C | 24D | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 24 | 24 | 24 | 24 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 12 | 2 | 2 | 84 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | SD16 | D12 | D14 | C24⋊C2 | C7⋊D4 | S3×D7 | D4.D7 | C7⋊D12 | D12.D7 |
| kernel | D12.D7 | C3×C7⋊C8 | C7×D12 | Dic42 | C7⋊C8 | C42 | C28 | D12 | C21 | C14 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 3 | 3 | 3 | 6 |
Matrix representation of D12.D7 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 15 | 322 |
| 0 | 0 | 15 | 30 |
| 336 | 0 | 0 | 0 |
| 71 | 1 | 0 | 0 |
| 0 | 0 | 15 | 30 |
| 0 | 0 | 15 | 322 |
| 8 | 0 | 0 | 0 |
| 247 | 295 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 63 | 35 | 0 | 0 |
| 31 | 274 | 0 | 0 |
| 0 | 0 | 313 | 93 |
| 0 | 0 | 244 | 220 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,15,15,0,0,322,30],[336,71,0,0,0,1,0,0,0,0,15,15,0,0,30,322],[8,247,0,0,0,295,0,0,0,0,1,0,0,0,0,1],[63,31,0,0,35,274,0,0,0,0,313,244,0,0,93,220] >;
D12.D7 in GAP, Magma, Sage, TeX
D_{12}.D_7 % in TeX
G:=Group("D12.D7"); // GroupNames label
G:=SmallGroup(336,36);
// by ID
G=gap.SmallGroup(336,36);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,169,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^7=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations
Export