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G = D7×C24order 336 = 24·3·7

Direct product of C24 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×C24, C1688C2, C5611C6, D14.4C12, C12.56D14, C84.64C22, Dic7.4C12, C7⋊C813C6, C74(C2×C24), C215(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

C1C7 — D7×C24
C1C7C14C28C84C12×D7 — D7×C24
C7 — D7×C24
C1C24

Generators and relations for D7×C24
 G = < a,b,c | a24=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C4
7C6
7C6
7C8
7C2×C4
7C12
7C2×C6
7C2×C8
7C24
7C2×C12
7C2×C24

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)
(1 113 55 85 46 148 124)(2 114 56 86 47 149 125)(3 115 57 87 48 150 126)(4 116 58 88 25 151 127)(5 117 59 89 26 152 128)(6 118 60 90 27 153 129)(7 119 61 91 28 154 130)(8 120 62 92 29 155 131)(9 97 63 93 30 156 132)(10 98 64 94 31 157 133)(11 99 65 95 32 158 134)(12 100 66 96 33 159 135)(13 101 67 73 34 160 136)(14 102 68 74 35 161 137)(15 103 69 75 36 162 138)(16 104 70 76 37 163 139)(17 105 71 77 38 164 140)(18 106 72 78 39 165 141)(19 107 49 79 40 166 142)(20 108 50 80 41 167 143)(21 109 51 81 42 168 144)(22 110 52 82 43 145 121)(23 111 53 83 44 146 122)(24 112 54 84 45 147 123)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 121)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 70)(26 71)(27 72)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(97 168)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)

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), (1,113,55,85,46,148,124)(2,114,56,86,47,149,125)(3,115,57,87,48,150,126)(4,116,58,88,25,151,127)(5,117,59,89,26,152,128)(6,118,60,90,27,153,129)(7,119,61,91,28,154,130)(8,120,62,92,29,155,131)(9,97,63,93,30,156,132)(10,98,64,94,31,157,133)(11,99,65,95,32,158,134)(12,100,66,96,33,159,135)(13,101,67,73,34,160,136)(14,102,68,74,35,161,137)(15,103,69,75,36,162,138)(16,104,70,76,37,163,139)(17,105,71,77,38,164,140)(18,106,72,78,39,165,141)(19,107,49,79,40,166,142)(20,108,50,80,41,167,143)(21,109,51,81,42,168,144)(22,110,52,82,43,145,121)(23,111,53,83,44,146,122)(24,112,54,84,45,147,123), (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,70)(26,71)(27,72)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(97,168)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)>;

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), (1,113,55,85,46,148,124)(2,114,56,86,47,149,125)(3,115,57,87,48,150,126)(4,116,58,88,25,151,127)(5,117,59,89,26,152,128)(6,118,60,90,27,153,129)(7,119,61,91,28,154,130)(8,120,62,92,29,155,131)(9,97,63,93,30,156,132)(10,98,64,94,31,157,133)(11,99,65,95,32,158,134)(12,100,66,96,33,159,135)(13,101,67,73,34,160,136)(14,102,68,74,35,161,137)(15,103,69,75,36,162,138)(16,104,70,76,37,163,139)(17,105,71,77,38,164,140)(18,106,72,78,39,165,141)(19,107,49,79,40,166,142)(20,108,50,80,41,167,143)(21,109,51,81,42,168,144)(22,110,52,82,43,145,121)(23,111,53,83,44,146,122)(24,112,54,84,45,147,123), (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,70)(26,71)(27,72)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(97,168)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167) );

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)], [(1,113,55,85,46,148,124),(2,114,56,86,47,149,125),(3,115,57,87,48,150,126),(4,116,58,88,25,151,127),(5,117,59,89,26,152,128),(6,118,60,90,27,153,129),(7,119,61,91,28,154,130),(8,120,62,92,29,155,131),(9,97,63,93,30,156,132),(10,98,64,94,31,157,133),(11,99,65,95,32,158,134),(12,100,66,96,33,159,135),(13,101,67,73,34,160,136),(14,102,68,74,35,161,137),(15,103,69,75,36,162,138),(16,104,70,76,37,163,139),(17,105,71,77,38,164,140),(18,106,72,78,39,165,141),(19,107,49,79,40,166,142),(20,108,50,80,41,167,143),(21,109,51,81,42,168,144),(22,110,52,82,43,145,121),(23,111,53,83,44,146,122),(24,112,54,84,45,147,123)], [(1,136),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,144),(10,121),(11,122),(12,123),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,70),(26,71),(27,72),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(97,168),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(113,160),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167)])

120 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F7A7B7C8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H14A14B14C21A···21F24A···24H24I···24P28A···28F42A···42F56A···56L84A···84L168A···168X
order122233444466666677788888888121212121212121214141421···2124···2424···2428···2842···4256···5684···84168···168
size117711117711777722211117777111177772222···21···17···72···22···22···22···22···2

120 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24D7D14C3×D7C4×D7C6×D7C8×D7C12×D7D7×C24
kernelD7×C24C3×C7⋊C8C168C12×D7C8×D7C3×Dic7C6×D7C7⋊C8C56C4×D7C3×D7Dic7D14D7C24C12C8C6C4C3C2C1
# reps11112222228441633666121224

Matrix representation of D7×C24 in GL3(𝔽337) generated by

28300
0850
0085
,
100
0336145
0336144
,
100
034159
0228303
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

Export

Subgroup lattice of D7×C24 in TeX

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