direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic14, C21⋊3Q8, C28.5C6, C84.3C2, C12.3D7, C6.13D14, Dic7.2C6, C42.13C22, C4.(C3×D7), C7⋊3(C3×Q8), C2.3(C6×D7), C14.9(C2×C6), (C3×Dic7).2C2, SmallGroup(168,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic14
G = < a,b,c | a3=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic14
►Regular action on 168 points
Generators in S168
(1 112 121)(2 85 122)(3 86 123)(4 87 124)(5 88 125)(6 89 126)(7 90 127)(8 91 128)(9 92 129)(10 93 130)(11 94 131)(12 95 132)(13 96 133)(14 97 134)(15 98 135)(16 99 136)(17 100 137)(18 101 138)(19 102 139)(20 103 140)(21 104 113)(22 105 114)(23 106 115)(24 107 116)(25 108 117)(26 109 118)(27 110 119)(28 111 120)(29 156 57)(30 157 58)(31 158 59)(32 159 60)(33 160 61)(34 161 62)(35 162 63)(36 163 64)(37 164 65)(38 165 66)(39 166 67)(40 167 68)(41 168 69)(42 141 70)(43 142 71)(44 143 72)(45 144 73)(46 145 74)(47 146 75)(48 147 76)(49 148 77)(50 149 78)(51 150 79)(52 151 80)(53 152 81)(54 153 82)(55 154 83)(56 155 84)
(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 44 15 30)(2 43 16 29)(3 42 17 56)(4 41 18 55)(5 40 19 54)(6 39 20 53)(7 38 21 52)(8 37 22 51)(9 36 23 50)(10 35 24 49)(11 34 25 48)(12 33 26 47)(13 32 27 46)(14 31 28 45)(57 122 71 136)(58 121 72 135)(59 120 73 134)(60 119 74 133)(61 118 75 132)(62 117 76 131)(63 116 77 130)(64 115 78 129)(65 114 79 128)(66 113 80 127)(67 140 81 126)(68 139 82 125)(69 138 83 124)(70 137 84 123)(85 142 99 156)(86 141 100 155)(87 168 101 154)(88 167 102 153)(89 166 103 152)(90 165 104 151)(91 164 105 150)(92 163 106 149)(93 162 107 148)(94 161 108 147)(95 160 109 146)(96 159 110 145)(97 158 111 144)(98 157 112 143)
G:=sub<Sym(168)| (1,112,121)(2,85,122)(3,86,123)(4,87,124)(5,88,125)(6,89,126)(7,90,127)(8,91,128)(9,92,129)(10,93,130)(11,94,131)(12,95,132)(13,96,133)(14,97,134)(15,98,135)(16,99,136)(17,100,137)(18,101,138)(19,102,139)(20,103,140)(21,104,113)(22,105,114)(23,106,115)(24,107,116)(25,108,117)(26,109,118)(27,110,119)(28,111,120)(29,156,57)(30,157,58)(31,158,59)(32,159,60)(33,160,61)(34,161,62)(35,162,63)(36,163,64)(37,164,65)(38,165,66)(39,166,67)(40,167,68)(41,168,69)(42,141,70)(43,142,71)(44,143,72)(45,144,73)(46,145,74)(47,146,75)(48,147,76)(49,148,77)(50,149,78)(51,150,79)(52,151,80)(53,152,81)(54,153,82)(55,154,83)(56,155,84), (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), (1,44,15,30)(2,43,16,29)(3,42,17,56)(4,41,18,55)(5,40,19,54)(6,39,20,53)(7,38,21,52)(8,37,22,51)(9,36,23,50)(10,35,24,49)(11,34,25,48)(12,33,26,47)(13,32,27,46)(14,31,28,45)(57,122,71,136)(58,121,72,135)(59,120,73,134)(60,119,74,133)(61,118,75,132)(62,117,76,131)(63,116,77,130)(64,115,78,129)(65,114,79,128)(66,113,80,127)(67,140,81,126)(68,139,82,125)(69,138,83,124)(70,137,84,123)(85,142,99,156)(86,141,100,155)(87,168,101,154)(88,167,102,153)(89,166,103,152)(90,165,104,151)(91,164,105,150)(92,163,106,149)(93,162,107,148)(94,161,108,147)(95,160,109,146)(96,159,110,145)(97,158,111,144)(98,157,112,143)>;
G:=Group( (1,112,121)(2,85,122)(3,86,123)(4,87,124)(5,88,125)(6,89,126)(7,90,127)(8,91,128)(9,92,129)(10,93,130)(11,94,131)(12,95,132)(13,96,133)(14,97,134)(15,98,135)(16,99,136)(17,100,137)(18,101,138)(19,102,139)(20,103,140)(21,104,113)(22,105,114)(23,106,115)(24,107,116)(25,108,117)(26,109,118)(27,110,119)(28,111,120)(29,156,57)(30,157,58)(31,158,59)(32,159,60)(33,160,61)(34,161,62)(35,162,63)(36,163,64)(37,164,65)(38,165,66)(39,166,67)(40,167,68)(41,168,69)(42,141,70)(43,142,71)(44,143,72)(45,144,73)(46,145,74)(47,146,75)(48,147,76)(49,148,77)(50,149,78)(51,150,79)(52,151,80)(53,152,81)(54,153,82)(55,154,83)(56,155,84), (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), (1,44,15,30)(2,43,16,29)(3,42,17,56)(4,41,18,55)(5,40,19,54)(6,39,20,53)(7,38,21,52)(8,37,22,51)(9,36,23,50)(10,35,24,49)(11,34,25,48)(12,33,26,47)(13,32,27,46)(14,31,28,45)(57,122,71,136)(58,121,72,135)(59,120,73,134)(60,119,74,133)(61,118,75,132)(62,117,76,131)(63,116,77,130)(64,115,78,129)(65,114,79,128)(66,113,80,127)(67,140,81,126)(68,139,82,125)(69,138,83,124)(70,137,84,123)(85,142,99,156)(86,141,100,155)(87,168,101,154)(88,167,102,153)(89,166,103,152)(90,165,104,151)(91,164,105,150)(92,163,106,149)(93,162,107,148)(94,161,108,147)(95,160,109,146)(96,159,110,145)(97,158,111,144)(98,157,112,143) );
G=PermutationGroup([[(1,112,121),(2,85,122),(3,86,123),(4,87,124),(5,88,125),(6,89,126),(7,90,127),(8,91,128),(9,92,129),(10,93,130),(11,94,131),(12,95,132),(13,96,133),(14,97,134),(15,98,135),(16,99,136),(17,100,137),(18,101,138),(19,102,139),(20,103,140),(21,104,113),(22,105,114),(23,106,115),(24,107,116),(25,108,117),(26,109,118),(27,110,119),(28,111,120),(29,156,57),(30,157,58),(31,158,59),(32,159,60),(33,160,61),(34,161,62),(35,162,63),(36,163,64),(37,164,65),(38,165,66),(39,166,67),(40,167,68),(41,168,69),(42,141,70),(43,142,71),(44,143,72),(45,144,73),(46,145,74),(47,146,75),(48,147,76),(49,148,77),(50,149,78),(51,150,79),(52,151,80),(53,152,81),(54,153,82),(55,154,83),(56,155,84)], [(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)], [(1,44,15,30),(2,43,16,29),(3,42,17,56),(4,41,18,55),(5,40,19,54),(6,39,20,53),(7,38,21,52),(8,37,22,51),(9,36,23,50),(10,35,24,49),(11,34,25,48),(12,33,26,47),(13,32,27,46),(14,31,28,45),(57,122,71,136),(58,121,72,135),(59,120,73,134),(60,119,74,133),(61,118,75,132),(62,117,76,131),(63,116,77,130),(64,115,78,129),(65,114,79,128),(66,113,80,127),(67,140,81,126),(68,139,82,125),(69,138,83,124),(70,137,84,123),(85,142,99,156),(86,141,100,155),(87,168,101,154),(88,167,102,153),(89,166,103,152),(90,165,104,151),(91,164,105,150),(92,163,106,149),(93,162,107,148),(94,161,108,147),(95,160,109,146),(96,159,110,145),(97,158,111,144),(98,157,112,143)]])
C3×Dic14 is a maximal subgroup of
C42.D4 C21⋊SD16 C21⋊Q16 C3⋊Dic28 D12⋊D7 D84⋊C2 D21⋊Q8 C3×Q8×D7
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | D7 | C3×Q8 | D14 | C3×D7 | Dic14 | C6×D7 | C3×Dic14 |
| kernel | C3×Dic14 | C3×Dic7 | C84 | Dic14 | Dic7 | C28 | C21 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C3×Dic14 ►in GL2(𝔽337) generated by
| 128 | 0 |
| 0 | 128 |
| 225 | 243 |
| 94 | 118 |
| 161 | 49 |
| 200 | 176 |
G:=sub<GL(2,GF(337))| [128,0,0,128],[225,94,243,118],[161,200,49,176] >;
C3×Dic14 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{14} % in TeX
G:=Group("C3xDic14"); // GroupNames label
G:=SmallGroup(168,24);
// by ID
G=gap.SmallGroup(168,24);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,3604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export