direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C24, C168⋊8C2, C56⋊11C6, D14.4C12, C12.56D14, C84.64C22, Dic7.4C12, C7⋊C8⋊13C6, C7⋊4(C2×C24), C21⋊5(C2×C8), (C6×D7).4C4, (C4×D7).6C6, C6.15(C4×D7), C2.1(C12×D7), C4.12(C6×D7), C28.36(C2×C6), C42.19(C2×C4), (C12×D7).6C2, C14.15(C2×C12), (C3×Dic7).4C4, (C3×C7⋊C8)⋊13C2, SmallGroup(336,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C24 |
Generators and relations for D7×C24
G = < a,b,c | a24=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C24
►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 75 150 62 99 122 33)(2 76 151 63 100 123 34)(3 77 152 64 101 124 35)(4 78 153 65 102 125 36)(5 79 154 66 103 126 37)(6 80 155 67 104 127 38)(7 81 156 68 105 128 39)(8 82 157 69 106 129 40)(9 83 158 70 107 130 41)(10 84 159 71 108 131 42)(11 85 160 72 109 132 43)(12 86 161 49 110 133 44)(13 87 162 50 111 134 45)(14 88 163 51 112 135 46)(15 89 164 52 113 136 47)(16 90 165 53 114 137 48)(17 91 166 54 115 138 25)(18 92 167 55 116 139 26)(19 93 168 56 117 140 27)(20 94 145 57 118 141 28)(21 95 146 58 119 142 29)(22 96 147 59 120 143 30)(23 73 148 60 97 144 31)(24 74 149 61 98 121 32)
(1 45)(2 46)(3 47)(4 48)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)
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), (1,75,150,62,99,122,33)(2,76,151,63,100,123,34)(3,77,152,64,101,124,35)(4,78,153,65,102,125,36)(5,79,154,66,103,126,37)(6,80,155,67,104,127,38)(7,81,156,68,105,128,39)(8,82,157,69,106,129,40)(9,83,158,70,107,130,41)(10,84,159,71,108,131,42)(11,85,160,72,109,132,43)(12,86,161,49,110,133,44)(13,87,162,50,111,134,45)(14,88,163,51,112,135,46)(15,89,164,52,113,136,47)(16,90,165,53,114,137,48)(17,91,166,54,115,138,25)(18,92,167,55,116,139,26)(19,93,168,56,117,140,27)(20,94,145,57,118,141,28)(21,95,146,58,119,142,29)(22,96,147,59,120,143,30)(23,73,148,60,97,144,31)(24,74,149,61,98,121,32), (1,45)(2,46)(3,47)(4,48)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159)>;
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), (1,75,150,62,99,122,33)(2,76,151,63,100,123,34)(3,77,152,64,101,124,35)(4,78,153,65,102,125,36)(5,79,154,66,103,126,37)(6,80,155,67,104,127,38)(7,81,156,68,105,128,39)(8,82,157,69,106,129,40)(9,83,158,70,107,130,41)(10,84,159,71,108,131,42)(11,85,160,72,109,132,43)(12,86,161,49,110,133,44)(13,87,162,50,111,134,45)(14,88,163,51,112,135,46)(15,89,164,52,113,136,47)(16,90,165,53,114,137,48)(17,91,166,54,115,138,25)(18,92,167,55,116,139,26)(19,93,168,56,117,140,27)(20,94,145,57,118,141,28)(21,95,146,58,119,142,29)(22,96,147,59,120,143,30)(23,73,148,60,97,144,31)(24,74,149,61,98,121,32), (1,45)(2,46)(3,47)(4,48)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159) );
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)], [(1,75,150,62,99,122,33),(2,76,151,63,100,123,34),(3,77,152,64,101,124,35),(4,78,153,65,102,125,36),(5,79,154,66,103,126,37),(6,80,155,67,104,127,38),(7,81,156,68,105,128,39),(8,82,157,69,106,129,40),(9,83,158,70,107,130,41),(10,84,159,71,108,131,42),(11,85,160,72,109,132,43),(12,86,161,49,110,133,44),(13,87,162,50,111,134,45),(14,88,163,51,112,135,46),(15,89,164,52,113,136,47),(16,90,165,53,114,137,48),(17,91,166,54,115,138,25),(18,92,167,55,116,139,26),(19,93,168,56,117,140,27),(20,94,145,57,118,141,28),(21,95,146,58,119,142,29),(22,96,147,59,120,143,30),(23,73,148,60,97,144,31),(24,74,149,61,98,121,32)], [(1,45),(2,46),(3,47),(4,48),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,121),(87,122),(88,123),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,157),(119,158),(120,159)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 14A | 14B | 14C | 21A | ··· | 21F | 24A | ··· | 24H | 24I | ··· | 24P | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | ··· | 24 | 24 | ··· | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | D7 | D14 | C3×D7 | C4×D7 | C6×D7 | C8×D7 | C12×D7 | D7×C24 |
| kernel | D7×C24 | C3×C7⋊C8 | C168 | C12×D7 | C8×D7 | C3×Dic7 | C6×D7 | C7⋊C8 | C56 | C4×D7 | C3×D7 | Dic7 | D14 | D7 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 3 | 3 | 6 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of D7×C24 ►in GL3(𝔽337) generated by
| 283 | 0 | 0 |
| 0 | 85 | 0 |
| 0 | 0 | 85 |
| 1 | 0 | 0 |
| 0 | 336 | 145 |
| 0 | 336 | 144 |
| 1 | 0 | 0 |
| 0 | 34 | 159 |
| 0 | 228 | 303 |
G:=sub<GL(3,GF(337))| [283,0,0,0,85,0,0,0,85],[1,0,0,0,336,336,0,145,144],[1,0,0,0,34,228,0,159,303] >;
D7×C24 in GAP, Magma, Sage, TeX
D_7\times C_{24} % in TeX
G:=Group("D7xC24"); // GroupNames label
G:=SmallGroup(336,58);
// by ID
G=gap.SmallGroup(336,58);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,79,69,10373]);
// Polycyclic
G:=Group<a,b,c|a^24=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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