direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C24⋊C2, C56⋊6S3, C24⋊2C14, C168⋊10C2, C21⋊9SD16, C42.28D4, C28.52D6, Dic6⋊1C14, D12.1C14, C14.13D12, C84.68C22, C8⋊2(S3×C7), C6.1(C7×D4), C3⋊1(C7×SD16), C4.8(S3×C14), C2.3(C7×D12), C12.8(C2×C14), (C7×Dic6)⋊7C2, (C7×D12).3C2, SmallGroup(336,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C24⋊C2
G = < a,b,c | a7=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Smallest permutation representation of C7×C24⋊C2
►On 168 points
Generators in S168
(1 52 129 29 84 147 111)(2 53 130 30 85 148 112)(3 54 131 31 86 149 113)(4 55 132 32 87 150 114)(5 56 133 33 88 151 115)(6 57 134 34 89 152 116)(7 58 135 35 90 153 117)(8 59 136 36 91 154 118)(9 60 137 37 92 155 119)(10 61 138 38 93 156 120)(11 62 139 39 94 157 97)(12 63 140 40 95 158 98)(13 64 141 41 96 159 99)(14 65 142 42 73 160 100)(15 66 143 43 74 161 101)(16 67 144 44 75 162 102)(17 68 121 45 76 163 103)(18 69 122 46 77 164 104)(19 70 123 47 78 165 105)(20 71 124 48 79 166 106)(21 72 125 25 80 167 107)(22 49 126 26 81 168 108)(23 50 127 27 82 145 109)(24 51 128 28 83 146 110)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 33)(26 44)(27 31)(28 42)(30 40)(32 38)(34 36)(35 47)(37 45)(39 43)(46 48)(49 67)(50 54)(51 65)(53 63)(55 61)(56 72)(57 59)(58 70)(60 68)(62 66)(69 71)(73 83)(74 94)(75 81)(76 92)(77 79)(78 90)(80 88)(82 86)(85 95)(87 93)(89 91)(97 101)(98 112)(100 110)(102 108)(103 119)(104 106)(105 117)(107 115)(109 113)(114 120)(116 118)(121 137)(122 124)(123 135)(125 133)(126 144)(127 131)(128 142)(130 140)(132 138)(134 136)(139 143)(145 149)(146 160)(148 158)(150 156)(151 167)(152 154)(153 165)(155 163)(157 161)(162 168)(164 166)
G:=sub<Sym(168)| (1,52,129,29,84,147,111)(2,53,130,30,85,148,112)(3,54,131,31,86,149,113)(4,55,132,32,87,150,114)(5,56,133,33,88,151,115)(6,57,134,34,89,152,116)(7,58,135,35,90,153,117)(8,59,136,36,91,154,118)(9,60,137,37,92,155,119)(10,61,138,38,93,156,120)(11,62,139,39,94,157,97)(12,63,140,40,95,158,98)(13,64,141,41,96,159,99)(14,65,142,42,73,160,100)(15,66,143,43,74,161,101)(16,67,144,44,75,162,102)(17,68,121,45,76,163,103)(18,69,122,46,77,164,104)(19,70,123,47,78,165,105)(20,71,124,48,79,166,106)(21,72,125,25,80,167,107)(22,49,126,26,81,168,108)(23,50,127,27,82,145,109)(24,51,128,28,83,146,110), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,33)(26,44)(27,31)(28,42)(30,40)(32,38)(34,36)(35,47)(37,45)(39,43)(46,48)(49,67)(50,54)(51,65)(53,63)(55,61)(56,72)(57,59)(58,70)(60,68)(62,66)(69,71)(73,83)(74,94)(75,81)(76,92)(77,79)(78,90)(80,88)(82,86)(85,95)(87,93)(89,91)(97,101)(98,112)(100,110)(102,108)(103,119)(104,106)(105,117)(107,115)(109,113)(114,120)(116,118)(121,137)(122,124)(123,135)(125,133)(126,144)(127,131)(128,142)(130,140)(132,138)(134,136)(139,143)(145,149)(146,160)(148,158)(150,156)(151,167)(152,154)(153,165)(155,163)(157,161)(162,168)(164,166)>;
G:=Group( (1,52,129,29,84,147,111)(2,53,130,30,85,148,112)(3,54,131,31,86,149,113)(4,55,132,32,87,150,114)(5,56,133,33,88,151,115)(6,57,134,34,89,152,116)(7,58,135,35,90,153,117)(8,59,136,36,91,154,118)(9,60,137,37,92,155,119)(10,61,138,38,93,156,120)(11,62,139,39,94,157,97)(12,63,140,40,95,158,98)(13,64,141,41,96,159,99)(14,65,142,42,73,160,100)(15,66,143,43,74,161,101)(16,67,144,44,75,162,102)(17,68,121,45,76,163,103)(18,69,122,46,77,164,104)(19,70,123,47,78,165,105)(20,71,124,48,79,166,106)(21,72,125,25,80,167,107)(22,49,126,26,81,168,108)(23,50,127,27,82,145,109)(24,51,128,28,83,146,110), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,33)(26,44)(27,31)(28,42)(30,40)(32,38)(34,36)(35,47)(37,45)(39,43)(46,48)(49,67)(50,54)(51,65)(53,63)(55,61)(56,72)(57,59)(58,70)(60,68)(62,66)(69,71)(73,83)(74,94)(75,81)(76,92)(77,79)(78,90)(80,88)(82,86)(85,95)(87,93)(89,91)(97,101)(98,112)(100,110)(102,108)(103,119)(104,106)(105,117)(107,115)(109,113)(114,120)(116,118)(121,137)(122,124)(123,135)(125,133)(126,144)(127,131)(128,142)(130,140)(132,138)(134,136)(139,143)(145,149)(146,160)(148,158)(150,156)(151,167)(152,154)(153,165)(155,163)(157,161)(162,168)(164,166) );
G=PermutationGroup([[(1,52,129,29,84,147,111),(2,53,130,30,85,148,112),(3,54,131,31,86,149,113),(4,55,132,32,87,150,114),(5,56,133,33,88,151,115),(6,57,134,34,89,152,116),(7,58,135,35,90,153,117),(8,59,136,36,91,154,118),(9,60,137,37,92,155,119),(10,61,138,38,93,156,120),(11,62,139,39,94,157,97),(12,63,140,40,95,158,98),(13,64,141,41,96,159,99),(14,65,142,42,73,160,100),(15,66,143,43,74,161,101),(16,67,144,44,75,162,102),(17,68,121,45,76,163,103),(18,69,122,46,77,164,104),(19,70,123,47,78,165,105),(20,71,124,48,79,166,106),(21,72,125,25,80,167,107),(22,49,126,26,81,168,108),(23,50,127,27,82,145,109),(24,51,128,28,83,146,110)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,33),(26,44),(27,31),(28,42),(30,40),(32,38),(34,36),(35,47),(37,45),(39,43),(46,48),(49,67),(50,54),(51,65),(53,63),(55,61),(56,72),(57,59),(58,70),(60,68),(62,66),(69,71),(73,83),(74,94),(75,81),(76,92),(77,79),(78,90),(80,88),(82,86),(85,95),(87,93),(89,91),(97,101),(98,112),(100,110),(102,108),(103,119),(104,106),(105,117),(107,115),(109,113),(114,120),(116,118),(121,137),(122,124),(123,135),(125,133),(126,144),(127,131),(128,142),(130,140),(132,138),(134,136),(139,143),(145,149),(146,160),(148,158),(150,156),(151,167),(152,154),(153,165),(155,163),(157,161),(162,168),(164,166)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6 | 7A | ··· | 7F | 8A | 8B | 12A | 12B | 14A | ··· | 14F | 14G | ··· | 14L | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 28G | ··· | 28L | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 7 | ··· | 7 | 8 | 8 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 12 | 2 | 2 | 12 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | S3 | D4 | D6 | SD16 | D12 | S3×C7 | C24⋊C2 | C7×D4 | S3×C14 | C7×SD16 | C7×D12 | C7×C24⋊C2 |
| kernel | C7×C24⋊C2 | C168 | C7×Dic6 | C7×D12 | C24⋊C2 | C24 | Dic6 | D12 | C56 | C42 | C28 | C21 | C14 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 2 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C7×C24⋊C2 ►in GL4(𝔽337) generated by
| 175 | 0 | 0 | 0 |
| 0 | 175 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 336 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 294 |
| 0 | 0 | 47 | 141 |
| 1 | 336 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 224 | 336 |
G:=sub<GL(4,GF(337))| [175,0,0,0,0,175,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,336,0,0,0,0,0,0,47,0,0,294,141],[1,0,0,0,336,336,0,0,0,0,1,224,0,0,0,336] >;
C7×C24⋊C2 in GAP, Magma, Sage, TeX
C_7\times C_{24}\rtimes C_2 % in TeX
G:=Group("C7xC24:C2"); // GroupNames label
G:=SmallGroup(336,76);
// by ID
G=gap.SmallGroup(336,76);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,175,2019,69,8069]);
// 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^11>;
// generators/relations
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