direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×D24, C21⋊6D8, C56⋊5S3, C24⋊1C14, C168⋊7C2, D12⋊1C14, C28.53D6, C42.29D4, C14.14D12, C84.69C22, C3⋊1(C7×D8), C8⋊1(S3×C7), C6.2(C7×D4), (C7×D12)⋊7C2, C4.9(S3×C14), C2.4(C7×D12), C12.9(C2×C14), SmallGroup(336,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D24
G = < a,b,c | a7=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D24
►On 168 points
Generators in S168
(1 122 34 107 88 66 156)(2 123 35 108 89 67 157)(3 124 36 109 90 68 158)(4 125 37 110 91 69 159)(5 126 38 111 92 70 160)(6 127 39 112 93 71 161)(7 128 40 113 94 72 162)(8 129 41 114 95 49 163)(9 130 42 115 96 50 164)(10 131 43 116 73 51 165)(11 132 44 117 74 52 166)(12 133 45 118 75 53 167)(13 134 46 119 76 54 168)(14 135 47 120 77 55 145)(15 136 48 97 78 56 146)(16 137 25 98 79 57 147)(17 138 26 99 80 58 148)(18 139 27 100 81 59 149)(19 140 28 101 82 60 150)(20 141 29 102 83 61 151)(21 142 30 103 84 62 152)(22 143 31 104 85 63 153)(23 144 32 105 86 64 154)(24 121 33 106 87 65 155)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(43 48)(44 47)(45 46)(49 58)(50 57)(51 56)(52 55)(53 54)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(73 78)(74 77)(75 76)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(117 120)(118 119)(121 122)(123 144)(124 143)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)(167 168)
G:=sub<Sym(168)| (1,122,34,107,88,66,156)(2,123,35,108,89,67,157)(3,124,36,109,90,68,158)(4,125,37,110,91,69,159)(5,126,38,111,92,70,160)(6,127,39,112,93,71,161)(7,128,40,113,94,72,162)(8,129,41,114,95,49,163)(9,130,42,115,96,50,164)(10,131,43,116,73,51,165)(11,132,44,117,74,52,166)(12,133,45,118,75,53,167)(13,134,46,119,76,54,168)(14,135,47,120,77,55,145)(15,136,48,97,78,56,146)(16,137,25,98,79,57,147)(17,138,26,99,80,58,148)(18,139,27,100,81,59,149)(19,140,28,101,82,60,150)(20,141,29,102,83,61,151)(21,142,30,103,84,62,152)(22,143,31,104,85,63,153)(23,144,32,105,86,64,154)(24,121,33,106,87,65,155), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(43,48)(44,47)(45,46)(49,58)(50,57)(51,56)(52,55)(53,54)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(73,78)(74,77)(75,76)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(117,120)(118,119)(121,122)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(167,168)>;
G:=Group( (1,122,34,107,88,66,156)(2,123,35,108,89,67,157)(3,124,36,109,90,68,158)(4,125,37,110,91,69,159)(5,126,38,111,92,70,160)(6,127,39,112,93,71,161)(7,128,40,113,94,72,162)(8,129,41,114,95,49,163)(9,130,42,115,96,50,164)(10,131,43,116,73,51,165)(11,132,44,117,74,52,166)(12,133,45,118,75,53,167)(13,134,46,119,76,54,168)(14,135,47,120,77,55,145)(15,136,48,97,78,56,146)(16,137,25,98,79,57,147)(17,138,26,99,80,58,148)(18,139,27,100,81,59,149)(19,140,28,101,82,60,150)(20,141,29,102,83,61,151)(21,142,30,103,84,62,152)(22,143,31,104,85,63,153)(23,144,32,105,86,64,154)(24,121,33,106,87,65,155), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(43,48)(44,47)(45,46)(49,58)(50,57)(51,56)(52,55)(53,54)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(73,78)(74,77)(75,76)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(117,120)(118,119)(121,122)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(167,168) );
G=PermutationGroup([[(1,122,34,107,88,66,156),(2,123,35,108,89,67,157),(3,124,36,109,90,68,158),(4,125,37,110,91,69,159),(5,126,38,111,92,70,160),(6,127,39,112,93,71,161),(7,128,40,113,94,72,162),(8,129,41,114,95,49,163),(9,130,42,115,96,50,164),(10,131,43,116,73,51,165),(11,132,44,117,74,52,166),(12,133,45,118,75,53,167),(13,134,46,119,76,54,168),(14,135,47,120,77,55,145),(15,136,48,97,78,56,146),(16,137,25,98,79,57,147),(17,138,26,99,80,58,148),(18,139,27,100,81,59,149),(19,140,28,101,82,60,150),(20,141,29,102,83,61,151),(21,142,30,103,84,62,152),(22,143,31,104,85,63,153),(23,144,32,105,86,64,154),(24,121,33,106,87,65,155)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(43,48),(44,47),(45,46),(49,58),(50,57),(51,56),(52,55),(53,54),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(73,78),(74,77),(75,76),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(117,120),(118,119),(121,122),(123,144),(124,143),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(145,166),(146,165),(147,164),(148,163),(149,162),(150,161),(151,160),(152,159),(153,158),(154,157),(155,156),(167,168)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 7A | ··· | 7F | 8A | 8B | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14R | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | S3 | D4 | D6 | D8 | D12 | S3×C7 | D24 | C7×D4 | S3×C14 | C7×D8 | C7×D12 | C7×D24 |
| kernel | C7×D24 | C168 | C7×D12 | D24 | C24 | D12 | C56 | C42 | C28 | C21 | C14 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 1 | 1 | 2 | 2 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C7×D24 ►in GL2(𝔽337) generated by
| 295 | 0 |
| 0 | 295 |
| 284 | 129 |
| 208 | 155 |
| 208 | 155 |
| 284 | 129 |
G:=sub<GL(2,GF(337))| [295,0,0,295],[284,208,129,155],[208,284,155,129] >;
C7×D24 in GAP, Magma, Sage, TeX
C_7\times D_{24} % in TeX
G:=Group("C7xD24"); // GroupNames label
G:=SmallGroup(336,77);
// by ID
G=gap.SmallGroup(336,77);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,511,2019,69,8069]);
// Polycyclic
G:=Group<a,b,c|a^7=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export