direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic7, C42⋊2C4, C14⋊3C12, C6.16D14, C42.16C22, C7⋊5(C2×C12), C21⋊8(C2×C4), (C2×C6).2D7, C2.2(C6×D7), C22.(C3×D7), (C2×C42).2C2, (C2×C14).3C6, C14.12(C2×C6), SmallGroup(168,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C6×Dic7 |
Generators and relations for C6×Dic7
G = < a,b,c | a6=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic7
►Regular action on 168 points
Generators in S168
(1 57 37 55 23 75)(2 58 38 56 24 76)(3 59 39 43 25 77)(4 60 40 44 26 78)(5 61 41 45 27 79)(6 62 42 46 28 80)(7 63 29 47 15 81)(8 64 30 48 16 82)(9 65 31 49 17 83)(10 66 32 50 18 84)(11 67 33 51 19 71)(12 68 34 52 20 72)(13 69 35 53 21 73)(14 70 36 54 22 74)(85 148 113 127 99 155)(86 149 114 128 100 156)(87 150 115 129 101 157)(88 151 116 130 102 158)(89 152 117 131 103 159)(90 153 118 132 104 160)(91 154 119 133 105 161)(92 141 120 134 106 162)(93 142 121 135 107 163)(94 143 122 136 108 164)(95 144 123 137 109 165)(96 145 124 138 110 166)(97 146 125 139 111 167)(98 147 126 140 112 168)
(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 92 8 85)(2 91 9 98)(3 90 10 97)(4 89 11 96)(5 88 12 95)(6 87 13 94)(7 86 14 93)(15 100 22 107)(16 99 23 106)(17 112 24 105)(18 111 25 104)(19 110 26 103)(20 109 27 102)(21 108 28 101)(29 114 36 121)(30 113 37 120)(31 126 38 119)(32 125 39 118)(33 124 40 117)(34 123 41 116)(35 122 42 115)(43 132 50 139)(44 131 51 138)(45 130 52 137)(46 129 53 136)(47 128 54 135)(48 127 55 134)(49 140 56 133)(57 141 64 148)(58 154 65 147)(59 153 66 146)(60 152 67 145)(61 151 68 144)(62 150 69 143)(63 149 70 142)(71 166 78 159)(72 165 79 158)(73 164 80 157)(74 163 81 156)(75 162 82 155)(76 161 83 168)(77 160 84 167)
G:=sub<Sym(168)| (1,57,37,55,23,75)(2,58,38,56,24,76)(3,59,39,43,25,77)(4,60,40,44,26,78)(5,61,41,45,27,79)(6,62,42,46,28,80)(7,63,29,47,15,81)(8,64,30,48,16,82)(9,65,31,49,17,83)(10,66,32,50,18,84)(11,67,33,51,19,71)(12,68,34,52,20,72)(13,69,35,53,21,73)(14,70,36,54,22,74)(85,148,113,127,99,155)(86,149,114,128,100,156)(87,150,115,129,101,157)(88,151,116,130,102,158)(89,152,117,131,103,159)(90,153,118,132,104,160)(91,154,119,133,105,161)(92,141,120,134,106,162)(93,142,121,135,107,163)(94,143,122,136,108,164)(95,144,123,137,109,165)(96,145,124,138,110,166)(97,146,125,139,111,167)(98,147,126,140,112,168), (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), (1,92,8,85)(2,91,9,98)(3,90,10,97)(4,89,11,96)(5,88,12,95)(6,87,13,94)(7,86,14,93)(15,100,22,107)(16,99,23,106)(17,112,24,105)(18,111,25,104)(19,110,26,103)(20,109,27,102)(21,108,28,101)(29,114,36,121)(30,113,37,120)(31,126,38,119)(32,125,39,118)(33,124,40,117)(34,123,41,116)(35,122,42,115)(43,132,50,139)(44,131,51,138)(45,130,52,137)(46,129,53,136)(47,128,54,135)(48,127,55,134)(49,140,56,133)(57,141,64,148)(58,154,65,147)(59,153,66,146)(60,152,67,145)(61,151,68,144)(62,150,69,143)(63,149,70,142)(71,166,78,159)(72,165,79,158)(73,164,80,157)(74,163,81,156)(75,162,82,155)(76,161,83,168)(77,160,84,167)>;
G:=Group( (1,57,37,55,23,75)(2,58,38,56,24,76)(3,59,39,43,25,77)(4,60,40,44,26,78)(5,61,41,45,27,79)(6,62,42,46,28,80)(7,63,29,47,15,81)(8,64,30,48,16,82)(9,65,31,49,17,83)(10,66,32,50,18,84)(11,67,33,51,19,71)(12,68,34,52,20,72)(13,69,35,53,21,73)(14,70,36,54,22,74)(85,148,113,127,99,155)(86,149,114,128,100,156)(87,150,115,129,101,157)(88,151,116,130,102,158)(89,152,117,131,103,159)(90,153,118,132,104,160)(91,154,119,133,105,161)(92,141,120,134,106,162)(93,142,121,135,107,163)(94,143,122,136,108,164)(95,144,123,137,109,165)(96,145,124,138,110,166)(97,146,125,139,111,167)(98,147,126,140,112,168), (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), (1,92,8,85)(2,91,9,98)(3,90,10,97)(4,89,11,96)(5,88,12,95)(6,87,13,94)(7,86,14,93)(15,100,22,107)(16,99,23,106)(17,112,24,105)(18,111,25,104)(19,110,26,103)(20,109,27,102)(21,108,28,101)(29,114,36,121)(30,113,37,120)(31,126,38,119)(32,125,39,118)(33,124,40,117)(34,123,41,116)(35,122,42,115)(43,132,50,139)(44,131,51,138)(45,130,52,137)(46,129,53,136)(47,128,54,135)(48,127,55,134)(49,140,56,133)(57,141,64,148)(58,154,65,147)(59,153,66,146)(60,152,67,145)(61,151,68,144)(62,150,69,143)(63,149,70,142)(71,166,78,159)(72,165,79,158)(73,164,80,157)(74,163,81,156)(75,162,82,155)(76,161,83,168)(77,160,84,167) );
G=PermutationGroup([[(1,57,37,55,23,75),(2,58,38,56,24,76),(3,59,39,43,25,77),(4,60,40,44,26,78),(5,61,41,45,27,79),(6,62,42,46,28,80),(7,63,29,47,15,81),(8,64,30,48,16,82),(9,65,31,49,17,83),(10,66,32,50,18,84),(11,67,33,51,19,71),(12,68,34,52,20,72),(13,69,35,53,21,73),(14,70,36,54,22,74),(85,148,113,127,99,155),(86,149,114,128,100,156),(87,150,115,129,101,157),(88,151,116,130,102,158),(89,152,117,131,103,159),(90,153,118,132,104,160),(91,154,119,133,105,161),(92,141,120,134,106,162),(93,142,121,135,107,163),(94,143,122,136,108,164),(95,144,123,137,109,165),(96,145,124,138,110,166),(97,146,125,139,111,167),(98,147,126,140,112,168)], [(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)], [(1,92,8,85),(2,91,9,98),(3,90,10,97),(4,89,11,96),(5,88,12,95),(6,87,13,94),(7,86,14,93),(15,100,22,107),(16,99,23,106),(17,112,24,105),(18,111,25,104),(19,110,26,103),(20,109,27,102),(21,108,28,101),(29,114,36,121),(30,113,37,120),(31,126,38,119),(32,125,39,118),(33,124,40,117),(34,123,41,116),(35,122,42,115),(43,132,50,139),(44,131,51,138),(45,130,52,137),(46,129,53,136),(47,128,54,135),(48,127,55,134),(49,140,56,133),(57,141,64,148),(58,154,65,147),(59,153,66,146),(60,152,67,145),(61,151,68,144),(62,150,69,143),(63,149,70,142),(71,166,78,159),(72,165,79,158),(73,164,80,157),(74,163,81,156),(75,162,82,155),(76,161,83,168),(77,160,84,167)]])
C6×Dic7 is a maximal subgroup of
D6⋊Dic7 D42⋊C4 C42.Q8 Dic21⋊C4 C14.Dic6 Dic3.D14 D7×C2×C12
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 7A | 7B | 7C | 12A | ··· | 12H | 14A | ··· | 14I | 21A | ··· | 21F | 42A | ··· | 42R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 7 | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D7 | Dic7 | D14 | C3×D7 | C3×Dic7 | C6×D7 |
| kernel | C6×Dic7 | C3×Dic7 | C2×C42 | C2×Dic7 | C42 | Dic7 | C2×C14 | C14 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 3 | 6 | 3 | 6 | 12 | 6 |
Matrix representation of C6×Dic7 ►in GL4(𝔽337) generated by
| 336 | 0 | 0 | 0 |
| 0 | 129 | 0 | 0 |
| 0 | 0 | 208 | 0 |
| 0 | 0 | 0 | 208 |
| 336 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 1 |
| 0 | 0 | 108 | 228 |
| 189 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 194 | 110 |
| 0 | 0 | 194 | 143 |
G:=sub<GL(4,GF(337))| [336,0,0,0,0,129,0,0,0,0,208,0,0,0,0,208],[336,0,0,0,0,1,0,0,0,0,336,108,0,0,1,228],[189,0,0,0,0,336,0,0,0,0,194,194,0,0,110,143] >;
C6×Dic7 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_7 % in TeX
G:=Group("C6xDic7"); // GroupNames label
G:=SmallGroup(168,27);
// by ID
G=gap.SmallGroup(168,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,60,3604]);
// Polycyclic
G:=Group<a,b,c|a^6=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export