metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊2D24, C21⋊3D8, D12⋊1D7, D84⋊8C2, C28.2D6, C42.3D4, C14.6D12, C12.22D14, C84.15C22, C7⋊C8⋊1S3, C3⋊1(D4⋊D7), C4.8(S3×D7), (C7×D12)⋊1C2, C6.1(C7⋊D4), C2.4(C7⋊D12), (C3×C7⋊C8)⋊1C2, SmallGroup(336,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊D24
G = < a,b,c | a7=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >
12C2
84C2
6C22
42C22
4S3
28S3
12D7
12C14
3D4
7C8
21D4
2D6
14D6
6D14
4D21
21D8
7D12
7C24
3D28
2D42
7D24
Smallest permutation representation of C7⋊D24
►On 168 points
Generators in S168
(1 37 56 92 108 148 133)(2 134 149 109 93 57 38)(3 39 58 94 110 150 135)(4 136 151 111 95 59 40)(5 41 60 96 112 152 137)(6 138 153 113 73 61 42)(7 43 62 74 114 154 139)(8 140 155 115 75 63 44)(9 45 64 76 116 156 141)(10 142 157 117 77 65 46)(11 47 66 78 118 158 143)(12 144 159 119 79 67 48)(13 25 68 80 120 160 121)(14 122 161 97 81 69 26)(15 27 70 82 98 162 123)(16 124 163 99 83 71 28)(17 29 72 84 100 164 125)(18 126 165 101 85 49 30)(19 31 50 86 102 166 127)(20 128 167 103 87 51 32)(21 33 52 88 104 168 129)(22 130 145 105 89 53 34)(23 35 54 90 106 146 131)(24 132 147 107 91 55 36)
(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 9)(2 8)(3 7)(4 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 144)(35 143)(36 142)(37 141)(38 140)(39 139)(40 138)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 163)(50 162)(51 161)(52 160)(53 159)(54 158)(55 157)(56 156)(57 155)(58 154)(59 153)(60 152)(61 151)(62 150)(63 149)(64 148)(65 147)(66 146)(67 145)(68 168)(69 167)(70 166)(71 165)(72 164)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)
G:=sub<Sym(168)| (1,37,56,92,108,148,133)(2,134,149,109,93,57,38)(3,39,58,94,110,150,135)(4,136,151,111,95,59,40)(5,41,60,96,112,152,137)(6,138,153,113,73,61,42)(7,43,62,74,114,154,139)(8,140,155,115,75,63,44)(9,45,64,76,116,156,141)(10,142,157,117,77,65,46)(11,47,66,78,118,158,143)(12,144,159,119,79,67,48)(13,25,68,80,120,160,121)(14,122,161,97,81,69,26)(15,27,70,82,98,162,123)(16,124,163,99,83,71,28)(17,29,72,84,100,164,125)(18,126,165,101,85,49,30)(19,31,50,86,102,166,127)(20,128,167,103,87,51,32)(21,33,52,88,104,168,129)(22,130,145,105,89,53,34)(23,35,54,90,106,146,131)(24,132,147,107,91,55,36), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,144)(35,143)(36,142)(37,141)(38,140)(39,139)(40,138)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,163)(50,162)(51,161)(52,160)(53,159)(54,158)(55,157)(56,156)(57,155)(58,154)(59,153)(60,152)(61,151)(62,150)(63,149)(64,148)(65,147)(66,146)(67,145)(68,168)(69,167)(70,166)(71,165)(72,164)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)>;
G:=Group( (1,37,56,92,108,148,133)(2,134,149,109,93,57,38)(3,39,58,94,110,150,135)(4,136,151,111,95,59,40)(5,41,60,96,112,152,137)(6,138,153,113,73,61,42)(7,43,62,74,114,154,139)(8,140,155,115,75,63,44)(9,45,64,76,116,156,141)(10,142,157,117,77,65,46)(11,47,66,78,118,158,143)(12,144,159,119,79,67,48)(13,25,68,80,120,160,121)(14,122,161,97,81,69,26)(15,27,70,82,98,162,123)(16,124,163,99,83,71,28)(17,29,72,84,100,164,125)(18,126,165,101,85,49,30)(19,31,50,86,102,166,127)(20,128,167,103,87,51,32)(21,33,52,88,104,168,129)(22,130,145,105,89,53,34)(23,35,54,90,106,146,131)(24,132,147,107,91,55,36), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,144)(35,143)(36,142)(37,141)(38,140)(39,139)(40,138)(41,137)(42,136)(43,135)(44,134)(45,133)(46,132)(47,131)(48,130)(49,163)(50,162)(51,161)(52,160)(53,159)(54,158)(55,157)(56,156)(57,155)(58,154)(59,153)(60,152)(61,151)(62,150)(63,149)(64,148)(65,147)(66,146)(67,145)(68,168)(69,167)(70,166)(71,165)(72,164)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112) );
G=PermutationGroup([[(1,37,56,92,108,148,133),(2,134,149,109,93,57,38),(3,39,58,94,110,150,135),(4,136,151,111,95,59,40),(5,41,60,96,112,152,137),(6,138,153,113,73,61,42),(7,43,62,74,114,154,139),(8,140,155,115,75,63,44),(9,45,64,76,116,156,141),(10,142,157,117,77,65,46),(11,47,66,78,118,158,143),(12,144,159,119,79,67,48),(13,25,68,80,120,160,121),(14,122,161,97,81,69,26),(15,27,70,82,98,162,123),(16,124,163,99,83,71,28),(17,29,72,84,100,164,125),(18,126,165,101,85,49,30),(19,31,50,86,102,166,127),(20,128,167,103,87,51,32),(21,33,52,88,104,168,129),(22,130,145,105,89,53,34),(23,35,54,90,106,146,131),(24,132,147,107,91,55,36)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,144),(35,143),(36,142),(37,141),(38,140),(39,139),(40,138),(41,137),(42,136),(43,135),(44,134),(45,133),(46,132),(47,131),(48,130),(49,163),(50,162),(51,161),(52,160),(53,159),(54,158),(55,157),(56,156),(57,155),(58,154),(59,153),(60,152),(61,151),(62,150),(63,149),(64,148),(65,147),(66,146),(67,145),(68,168),(69,167),(70,166),(71,165),(72,164),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 24A | 24B | 24C | 24D | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 24 | 24 | 24 | 24 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 12 | 84 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | D8 | D12 | D14 | D24 | C7⋊D4 | S3×D7 | D4⋊D7 | C7⋊D12 | C7⋊D24 |
| kernel | C7⋊D24 | C3×C7⋊C8 | C7×D12 | D84 | C7⋊C8 | C42 | C28 | D12 | C21 | C14 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 3 | 3 | 3 | 6 |
Matrix representation of C7⋊D24 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 227 |
| 76 | 263 | 0 | 0 |
| 242 | 26 | 0 | 0 |
| 0 | 0 | 45 | 292 |
| 0 | 0 | 60 | 292 |
| 1 | 36 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 33 | 33 |
| 0 | 0 | 110 | 304 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,0,336,0,0,1,227],[76,242,0,0,263,26,0,0,0,0,45,60,0,0,292,292],[1,0,0,0,36,336,0,0,0,0,33,110,0,0,33,304] >;
C7⋊D24 in GAP, Magma, Sage, TeX
C_7\rtimes D_{24} % in TeX
G:=Group("C7:D24"); // GroupNames label
G:=SmallGroup(336,31);
// by ID
G=gap.SmallGroup(336,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,73,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^7=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export