direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×Dic6, C21⋊4Q8, C84.5C2, C28.3S3, C12.1C14, C14.13D6, Dic3.C14, C42.18C22, C3⋊(C7×Q8), C4.(S3×C7), C2.3(S3×C14), C6.1(C2×C14), (C7×Dic3).2C2, SmallGroup(168,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Dic6
G = < a,b,c | a7=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Dic6
►Regular action on 168 points
Generators in S168
(1 141 16 65 52 159 44)(2 142 17 66 53 160 45)(3 143 18 67 54 161 46)(4 144 19 68 55 162 47)(5 133 20 69 56 163 48)(6 134 21 70 57 164 37)(7 135 22 71 58 165 38)(8 136 23 72 59 166 39)(9 137 24 61 60 167 40)(10 138 13 62 49 168 41)(11 139 14 63 50 157 42)(12 140 15 64 51 158 43)(25 145 110 92 122 83 97)(26 146 111 93 123 84 98)(27 147 112 94 124 73 99)(28 148 113 95 125 74 100)(29 149 114 96 126 75 101)(30 150 115 85 127 76 102)(31 151 116 86 128 77 103)(32 152 117 87 129 78 104)(33 153 118 88 130 79 105)(34 154 119 89 131 80 106)(35 155 120 90 132 81 107)(36 156 109 91 121 82 108)
(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 80 7 74)(2 79 8 73)(3 78 9 84)(4 77 10 83)(5 76 11 82)(6 75 12 81)(13 25 19 31)(14 36 20 30)(15 35 21 29)(16 34 22 28)(17 33 23 27)(18 32 24 26)(37 126 43 132)(38 125 44 131)(39 124 45 130)(40 123 46 129)(41 122 47 128)(42 121 48 127)(49 110 55 116)(50 109 56 115)(51 120 57 114)(52 119 58 113)(53 118 59 112)(54 117 60 111)(61 146 67 152)(62 145 68 151)(63 156 69 150)(64 155 70 149)(65 154 71 148)(66 153 72 147)(85 157 91 163)(86 168 92 162)(87 167 93 161)(88 166 94 160)(89 165 95 159)(90 164 96 158)(97 144 103 138)(98 143 104 137)(99 142 105 136)(100 141 106 135)(101 140 107 134)(102 139 108 133)
G:=sub<Sym(168)| (1,141,16,65,52,159,44)(2,142,17,66,53,160,45)(3,143,18,67,54,161,46)(4,144,19,68,55,162,47)(5,133,20,69,56,163,48)(6,134,21,70,57,164,37)(7,135,22,71,58,165,38)(8,136,23,72,59,166,39)(9,137,24,61,60,167,40)(10,138,13,62,49,168,41)(11,139,14,63,50,157,42)(12,140,15,64,51,158,43)(25,145,110,92,122,83,97)(26,146,111,93,123,84,98)(27,147,112,94,124,73,99)(28,148,113,95,125,74,100)(29,149,114,96,126,75,101)(30,150,115,85,127,76,102)(31,151,116,86,128,77,103)(32,152,117,87,129,78,104)(33,153,118,88,130,79,105)(34,154,119,89,131,80,106)(35,155,120,90,132,81,107)(36,156,109,91,121,82,108), (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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,25,19,31)(14,36,20,30)(15,35,21,29)(16,34,22,28)(17,33,23,27)(18,32,24,26)(37,126,43,132)(38,125,44,131)(39,124,45,130)(40,123,46,129)(41,122,47,128)(42,121,48,127)(49,110,55,116)(50,109,56,115)(51,120,57,114)(52,119,58,113)(53,118,59,112)(54,117,60,111)(61,146,67,152)(62,145,68,151)(63,156,69,150)(64,155,70,149)(65,154,71,148)(66,153,72,147)(85,157,91,163)(86,168,92,162)(87,167,93,161)(88,166,94,160)(89,165,95,159)(90,164,96,158)(97,144,103,138)(98,143,104,137)(99,142,105,136)(100,141,106,135)(101,140,107,134)(102,139,108,133)>;
G:=Group( (1,141,16,65,52,159,44)(2,142,17,66,53,160,45)(3,143,18,67,54,161,46)(4,144,19,68,55,162,47)(5,133,20,69,56,163,48)(6,134,21,70,57,164,37)(7,135,22,71,58,165,38)(8,136,23,72,59,166,39)(9,137,24,61,60,167,40)(10,138,13,62,49,168,41)(11,139,14,63,50,157,42)(12,140,15,64,51,158,43)(25,145,110,92,122,83,97)(26,146,111,93,123,84,98)(27,147,112,94,124,73,99)(28,148,113,95,125,74,100)(29,149,114,96,126,75,101)(30,150,115,85,127,76,102)(31,151,116,86,128,77,103)(32,152,117,87,129,78,104)(33,153,118,88,130,79,105)(34,154,119,89,131,80,106)(35,155,120,90,132,81,107)(36,156,109,91,121,82,108), (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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,25,19,31)(14,36,20,30)(15,35,21,29)(16,34,22,28)(17,33,23,27)(18,32,24,26)(37,126,43,132)(38,125,44,131)(39,124,45,130)(40,123,46,129)(41,122,47,128)(42,121,48,127)(49,110,55,116)(50,109,56,115)(51,120,57,114)(52,119,58,113)(53,118,59,112)(54,117,60,111)(61,146,67,152)(62,145,68,151)(63,156,69,150)(64,155,70,149)(65,154,71,148)(66,153,72,147)(85,157,91,163)(86,168,92,162)(87,167,93,161)(88,166,94,160)(89,165,95,159)(90,164,96,158)(97,144,103,138)(98,143,104,137)(99,142,105,136)(100,141,106,135)(101,140,107,134)(102,139,108,133) );
G=PermutationGroup([[(1,141,16,65,52,159,44),(2,142,17,66,53,160,45),(3,143,18,67,54,161,46),(4,144,19,68,55,162,47),(5,133,20,69,56,163,48),(6,134,21,70,57,164,37),(7,135,22,71,58,165,38),(8,136,23,72,59,166,39),(9,137,24,61,60,167,40),(10,138,13,62,49,168,41),(11,139,14,63,50,157,42),(12,140,15,64,51,158,43),(25,145,110,92,122,83,97),(26,146,111,93,123,84,98),(27,147,112,94,124,73,99),(28,148,113,95,125,74,100),(29,149,114,96,126,75,101),(30,150,115,85,127,76,102),(31,151,116,86,128,77,103),(32,152,117,87,129,78,104),(33,153,118,88,130,79,105),(34,154,119,89,131,80,106),(35,155,120,90,132,81,107),(36,156,109,91,121,82,108)], [(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)], [(1,80,7,74),(2,79,8,73),(3,78,9,84),(4,77,10,83),(5,76,11,82),(6,75,12,81),(13,25,19,31),(14,36,20,30),(15,35,21,29),(16,34,22,28),(17,33,23,27),(18,32,24,26),(37,126,43,132),(38,125,44,131),(39,124,45,130),(40,123,46,129),(41,122,47,128),(42,121,48,127),(49,110,55,116),(50,109,56,115),(51,120,57,114),(52,119,58,113),(53,118,59,112),(54,117,60,111),(61,146,67,152),(62,145,68,151),(63,156,69,150),(64,155,70,149),(65,154,71,148),(66,153,72,147),(85,157,91,163),(86,168,92,162),(87,167,93,161),(88,166,94,160),(89,165,95,159),(90,164,96,158),(97,144,103,138),(98,143,104,137),(99,142,105,136),(100,141,106,135),(101,140,107,134),(102,139,108,133)]])
C7×Dic6 is a maximal subgroup of
C28.D6 Dic6⋊D7 C21⋊Q16 C7⋊Dic12 D28⋊S3 D21⋊Q8 D14.D6 S3×C7×Q8
63 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 7A | ··· | 7F | 12A | 12B | 14A | ··· | 14F | 21A | ··· | 21F | 28A | ··· | 28F | 28G | ··· | 28R | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 7 | ··· | 7 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | S3 | Q8 | D6 | Dic6 | S3×C7 | C7×Q8 | S3×C14 | C7×Dic6 |
| kernel | C7×Dic6 | C7×Dic3 | C84 | Dic6 | Dic3 | C12 | C28 | C21 | C14 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 1 | 1 | 1 | 2 | 6 | 6 | 6 | 12 |
Matrix representation of C7×Dic6 ►in GL4(𝔽337) generated by
| 79 | 0 | 0 | 0 |
| 0 | 79 | 0 | 0 |
| 0 | 0 | 79 | 0 |
| 0 | 0 | 0 | 79 |
| 0 | 336 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 224 |
| 0 | 0 | 3 | 336 |
| 296 | 146 | 0 | 0 |
| 187 | 41 | 0 | 0 |
| 0 | 0 | 302 | 240 |
| 0 | 0 | 301 | 35 |
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,79,0,0,0,0,79],[0,1,0,0,336,1,0,0,0,0,1,3,0,0,224,336],[296,187,0,0,146,41,0,0,0,0,302,301,0,0,240,35] >;
C7×Dic6 in GAP, Magma, Sage, TeX
C_7\times {\rm Dic}_6 % in TeX
G:=Group("C7xDic6"); // GroupNames label
G:=SmallGroup(168,29);
// by ID
G=gap.SmallGroup(168,29);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-3,140,301,146,2804]);
// Polycyclic
G:=Group<a,b,c|a^7=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export